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Global existence and blowup of a localized problem with free boundary
Nonlinear Analysis 74 (2011) 2523–2533
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Nonlinear Analysis journal homepage: www.elsevier.com/locate/na
Global existence and blowup of a localized problem with free boundary✩ Peng Zhou, Jie Bao, Zhigui Lin ∗ School of Mathematical Science, Yangzhou University, Yangzhou 225002, PR China
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abstract
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Article history: Received 15 July 2010 Accepted 27 November 2010
This paper is concerned with a double fronts free boundary problem for the heat equation with a localized nonlinear reaction term. The local existence and uniqueness of the solution are given by applying the contraction mapping theorem. Then we present some conditions so that the solution blows up in finite time. Finally, the long-time behavior of the global solution is discussed. We show that the solution is global and fast if the initial data is small and that a global slow solution is possible when the initial data is suitably large. © 2010 Elsevier Ltd. All rights reserved.
MSC: 35K20 35R35 92B05 Keywords: Free boundary Blowup Global fast solution Global slow solution Localized
1. Introduction In this paper, we investigate the behavior of the positive solution u(t , x) of the following localized problem with double fronts free boundary
u − duxx = up (t , 0), t > 0, g (t ) < x < h(t ), t u(t , g (t )) = 0, g ′ (t ) = −µux (t , g (t )), t > 0, h′ (t ) = −µux (t , h(t )), t > 0, u(t , h(t )) = 0, g (0) = −h0 , h(0) = h0 , u(0, x) = u0 (x), −h0 ≤ x ≤ h0 ,
(1.1)
where both x = g (t ) and x = h(t ) are moving boundaries to be determined, h0 > 0, P > 1, d and µ are positive constants, and the initial function u0 satisfies
u0 ∈ C 2 ([−h0 , h0 ]), u0 (−h0 ) = u0 (h0 ) = 0,
and u0 > 0 in (−h0 , h0 ).
(1.2)
In [1,2], the authors considered a Stefan problem with nonlocal superlinear term, and exhibited an energy condition under which the solution would blow up in finite time. In addition, they proved that all global solutions are bounded and decay uniformly to 0. Moreover, they showed that the free boundary converges to a finite limit and the solution decays at an exponential rate, or the free boundary grows up to infinity and the decay rate is at most polynomial. For the localized equation ut − 1u = f (u(t , x0 )) ✩ The work is partially supported by PRC grant NSFC 11071209, 10801115 and also by NSF 10KJB110011 of Jiangsu Province.
∗
Corresponding author. E-mail address: [email protected] (Z. Lin).
0362-546X/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2010.11.047
(1.3)
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in a fixed domain, much work has been done to consider the long-time behavior of its solution. For example, the authors in [3] investigated this equation with Dirichlet and Neumann boundary conditions, respectively, as well as the corresponding Cauchy problem. They showed that under certain conditions the solution would cease to exist in finite time. Moreover, they proved that the set of all blowup points is the whole region, and estimated the growth rate of solutions near the blowup time. As we know, Eq. (1.3) describes some physical phenomena in which the nonlinear reaction in a dynamical system takes place only at a single (or sometimes several) site(s). As an example, the influence of defect structures on a catalytic surface can be modeled by a similar equation. One can find more detailed information about the physical derivation by consulting [4,5]. Recently, the free boundary problem has attracted much attention in many areas, for instance, the decrease of oxygen in a muscle in the vicinity of a clotted blood vessel [6], the American option pricing problem [7,8], wound healing [9], combustion under gravity conditions [10], tumor growth [11] and the dynamics of a population [12–16]. In this paper, we consider the localized problem with free boundary (1.1) and pay much attention to the blowup property of the solution and the long-time behavior of the global solutions. We will give some sufficient conditions ensuring the existence of a fast solution and a slow solution. Here, we say that the solution exists globally if Tmax = +∞, whereas if the solution ceases to exist for some finite time, that is, Tmax < +∞ and limt →Tmax ‖u(t , x)‖L∞ ([0,t ]×[g (t ),h(t )]) = +∞, we say that the solution blows up. If Tmax = ∞, limt →∞ h(t ) < ∞ and limt →∞ g (t ) < ∞, then the solution is called fast solution, since that the solution decays uniformly to 0 at an exponential rate, while if Tmax = ∞, limt →∞ h(t ) = ∞ and limt →∞ g (t ) = ∞, it is called slow solution, whose decay rate is at most polynomial; see [1,2] for details. The paper is organized as follows. First, local existence and uniqueness are exhibited in Section 2 as well as some preliminary results such as the maximum principle. Section 3 is devoted to the blowup property. Our result shows that the solution will blow up if the initial data is sufficiently large. Section 4 deals with the long-time behavior of global solutions, including the existence of a global fast solution and a global slow solution. 2. Local existence and uniqueness In this section, by using the contraction mapping theorem, we first give the following local existence and uniqueness result. And then we use the Hopf lemma to verify the monotonicity of the double free boundary fronts. Lastly, some preliminary results about the comparison principle are presented. Theorem 2.1. For any given u0 satisfying (1.2) and any α ∈ (0, 1), there is a T > 0 such that problem (1.1) admits a unique solution
(u, h, g ) ∈ C (1+α)/2,1+α (DT ) × C 1+α/2 ([0, T ]) × C 1+α/2 ([0, T ]); moreover,
‖u‖C (1+α)/2,1+α (DT ) + ‖h‖C 1+α/2 ([0,T ]) + ‖g ‖C 1+α/2 ([0,T ]) ≤ C ,
(2.1)
where DT = {(t , x) ∈ R2 : t ∈ [0, T ], x ∈ [g (t ), h(t )]}, C and T depend only on h0 , α and ‖u0 ‖C 2 ([−h0 ,h0 ]) . Proof. We first make a change of variable to straighten the free boundary. Let y=
2h0 x h(t ) − g (t )
−
h0 (h(t ) + g (t )) h(t ) − g (t )
,
v(t , y) = u(t , x).
Then problem (1.1) becomes
h(t ) + g (t ) p , t > 0, − h0 < y < h0 , v t , −h 0 t − Avy − Bvyy = v h(t ) − g (t ) 2h0 µ ∂v v = 0, h′ (t ) = − , t > 0, y = h0 , h(t ) − g (t ) ∂ y 2h0 µ ∂v v = 0, g ′ (t ) = − , t > 0, y = −h0 , h(t ) − g (t ) ∂ y h(0) = h0 , g (0) = −h0 , v(0, y) = v0 (y) := u0 (y), −h0 ≤ y ≤ h0 , h′ (t )−g ′ (t )
4h2 d
h′ (t )+g ′ (t )
(2.2)
where A = A(h, g , y) = y h(t )−g (t ) + h0 h(t )−g (t ) and B = B(h, g ) = (h(t )−0g (t ))2 . This transformation changes the free boundaries x = h(t ) and x = g (t ) to the fixed lines x = h0 and x = −h0 , respectively, but at the expense of making the equation more complex, as the coefficients in the first equation of (2.2) now include the unknown function h(t ) and g (t ). We denote k1 = −µv0′ (−h0 ) and k2 = −µv0′ (h0 ), and set
GT = g ∈ C 1 [0, T ] : g (0) = −h0 , g ′ (0) = k1 , ‖g ′ (t ) − k1 ‖ ⩽ 1 ,
HT = h ∈ C 1 [0, T ] : h(0) = h0 , h′ (0) = k2 , ‖h′ (t ) − k2 ‖ ⩽ 1 ,
P. Zhou et al. / Nonlinear Analysis 74 (2011) 2523–2533
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UT = v ∈ C ([0, T ] × [−h0 , h0 ]) : v(0, y) = v0 (y), ‖v − v0 ‖C ([0,T ]×[−h0 ,h0 ]) ⩽ 1 . It is not difficult to see that ΓT := UT × GT × HT is a complete metric space with the metric
D ((v1 , g1 , h1 ), (v2 , g2 , h2 )) = ‖v1 − v2 ‖C ([0,T ]×[−h0 ,h0 ]) + ‖g1′ − g2′ ‖C ([0,T ]) + ‖h′1 − h′2 ‖C ([0,T ]) . Let us notice that, for g1 , g2 ∈ GT , since g1 (0) = g2 (0) = −h0 , we have
‖g1 − g2 ‖C ([0,T ]) ≤ T ‖g1′ − g2′ ‖C ([0,T ]) .
(2.3)
Similarly we have that, for h1 , h2 ∈ HT ,
‖h1 − h2 ‖C ([0,T ]) ≤ T ‖h′1 − h′2 ‖C ([0,T ]) .
(2.4)
Next, by using standard L theory and the Sobolev imbedding theorem [17], we can obtain, for any (v, g , h) ∈ ΓT , the following initial boundary value problem: p
h+g p , −h 0 < y < h 0 , 0 < t < T , v˜ t − Av˜ y − Bv˜ yy = v t , −h0 h−g v˜ (t , h0 ) = v˜ (t , −h0 ) = 0, 0 < t < T , v˜ (0, y) = v0 (y) ⩾ 0, −h0 ⩽ y ⩽ h0
(2.5)
admits a unique solution v˜ ∈ C (1+α)/2,1+α ([0, T ] × [−h0 , h0 ]) and
‖˜v ‖C (1+α)/2,1+α ([0,T ]×[−h0 ,h0 ]) ⩽ C ‖˜v ‖W 1,2,p ([0,T ]×[−h0 ,h0 ]) ⩽ C1 ,
(2.6)
3 where A, B is defined as that in (2.2), p = 1−α , and C1 is a constant depending on α, h0 and ‖u0 ‖C 1+α [−h0 ,h0 ] . Now, we define s˜(t ) as the following:
s˜ = (2h0 ) − 4h0 µ 2
2
t
∫
v˜ y (τ , h0 ) − v˜ y (τ , −h0 )dτ ,
(2.7)
0
and we then have s˜(0) = 2h0 . Based on s˜(t ), we now define h˜ (t ) − h0 =
t
∫ 0
−2h0 µ v˜ y (τ , h0 )dτ s˜(τ )
(2.8)
−2h0 µ v˜ y (τ , −h0 )dτ . s˜(τ )
(2.9)
and g˜ (t ) + h0 =
t
∫ 0
According to (2.8) and (2.9), we can easily get h˜′ (t ) =
−2h0 µ v˜ y (t , h0 ) s˜(t )
g˜′ (t ) =
−2h0 µ v˜ y (t , −h0 ), s˜(t )
and
which also imply
(h˜ ′ (t ) − g˜ ′ (t ))˜s(t ) = s˜(t )˜s′ (t ) due to (2.7). Thus we have h˜ ′ (t ) − g˜ ′ (t ) = s˜′ (t ). Since h˜ (0) − g˜ (0) = s˜(0) = 2h0 , we get h˜ (t ) − g˜ (t ) = s˜(t ). Now, we obtain the following two equations: h˜′ (t ) =
−2h0 µ v˜ (t , h0 ) ˜h(t ) − g˜ (t ) y
(2.10)
g˜′ (t ) =
−2h0 µ v˜ (t , −h0 ), ˜h(t ) − g˜ (t ) y
(2.11)
and
with h˜ (0) = −˜g (0) = h0 , h˜ ′ (0) = k2 , and g˜ ′ (0) = k1 , and hence h˜ ′ (t ), g˜ ′ (t ) ∈ C α/2 ([0, T ]), with
‖h˜ ′ (t )‖C α/2 ([0,T ]) ⩽ C2 := µC1
(2.12)
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and
‖˜g ′ (t )‖C α/2 ([0,T ]) ⩽ C2 := µC1 .
(2.13)
Next we define a map F : ΓT −→ C ([0, T ] × [−h0 , h0 ]) × C 1 [0, T ] × C 1 [0, T ] by
F (v(t , y), g (t ), h(t )) = (˜v (t , y), g˜ (t ), h˜ (t )). It is obvious that (v(t , y), g (t ), h(t )) ∈ ΓT is a fixed point of F if and only if it solves (2.2). By (2.6), (2.12) and (2.13), we have
‖h˜ ′ − k2 ‖C ([0,T ]) ≤ ‖h˜ ′ ‖C α/2 ([0,T ]) T α/2 ≤ µC1 T α/2 , ‖˜g ′ − k1 ‖C ([0,T ]) ≤ ‖˜g ′ ‖C α/2 ([0,T ]) T α/2 ≤ µC1 T α/2 , ‖˜v − u0 ‖C ([0,T ]×[−h0 ,h0 ]) ≤ ‖˜v − u0 ‖C (1+α)/2,0 ([0,T ]×[−h0 ,h0 ]) T (1+α)/2 ≤ C1 T (1+α)/2 . −2/(1+α)
Therefore, if we take T ≤ min{(µC1 )−2/α , C1 }, then F maps ΓT into itself. To demonstrate that F is a contraction mapping over ΓT for T > 0 sufficiently small, we take (vi , gi , hi ) ∈ ΓT (i = 1, 2), and denote (v i , g i , hi ) = F (vi , gi , hi ). Then, it follows from (2.6), (2.12) and (2.13) that
‖v i ‖C (1+α)/2,1+α ([0,T ]×[−h0 ,h0 ]) ⩽ C ‖v i ‖W 1,2,p ([0,T ]×[−h0 ,h0 ]) ⩽ C1 , ‖g ′i (t )‖C α/2 [0,T ] ⩽ C2 ,
′
‖hi (t )‖C α/2 [0,T ] ⩽ C2 .
By setting W = v 1 − v 2 and performing some direct computation, we can deduce that W (t , y) satisfies Wt − A(h1 , g1 , y)Wy − B(h1 , g1 )Wyy = [A(h1 , g1 , y) − A(h2 , g2 , y)]v 2,y + [B(h1 , g1 ) − B(h2 , g2 )]v 2,yy h1 + g1 h2 + g2 + v1p t , −h0 − v2p t , −h0 , t > 0, h0 < y < h0 , h1 − g1 h2 − g2 W (t , −h0 ) = 0, W (0, y) = 0,
W ( t , h0 ) = 0 ,
t > 0,
−h 0 ≤ y ≤ h 0 .
p
Applying the L estimates for parabolic equations and Sobolev’s imbedding theorem, we obtain
‖v 1 − v 2 ‖C (1+α)/2,1+α ([0,T ]×[−h0 ,h0 ]) ≤ C3 (‖v1 − v2 ‖C ([0,T ]×[−h0 ,h0 ]) + ‖h1 − h2 ‖C 1 ([0,T ]) + ‖g1 − g2 ‖C 1 ([0,T ]) ),
(2.14)
where C3 depends on C1 and C2 , and the functions A, B and C are defined in (2.2). Taking the difference of the equations for h1 , h2 , g 1 and g 2 results in
′ ′ ‖h1 − h2 ‖C α/2 ([0,T ]) ≤ µ ‖w1,y − w2,y ‖C α/2,0 ([0,T ]×[−h0 ,h0 ])
(2.15)
‖g ′1 − g ′2 ‖C α/2 ([0,T ]) ≤ µ ‖w1,y − w2,y ‖C α/2,0 ([0,T ]×[−h0 ,h0 ]) .
(2.16)
and
Combining inequalities (2.14)–(2.16), and assuming that T ≤ 1, we get ′
′
‖v 1 − v 2 ‖C (1+α)/2,1+α ([0,T ]×[−h0 ,h0 ]) + ‖h1 − h2 ‖C α/2 ([0,T ]) + ‖g ′1 − g ′2 ‖C α/2 ([0,T ]) ≤ C4 (‖v1 − v2 ‖C ([0,T ]×[−h0 ,h0 ]) + ‖h′1 − h′2 ‖C [0,T ] + ‖g1′ − g2′ ‖C [0,T ] ) with C4 depending on C3 and µ. Hence, for
2/α 1 −2/(1+α) −2/α T := min 1, , (µC1 ) , C1 , 2C4
we have ′
′
‖v 1 − v 2 ‖C ([0,T ]×[−h0 ,h0 ]) + ‖h1 − h2 ‖C ([0,T ]) + ‖g ′1 − g ′2 ‖C ([0,T ]) ′
′
≤ T (1+α)/2 ‖v 1 − v 2 ‖C (1+α)/2,1+α ([0,T ]×[−h0 ,h0 ]) + T α/2 (‖h1 − h2 ‖C α/2 ([0,T ]) + ‖g ′1 − g ′2 ‖C α/2 ([0,T ]) ) ≤ C4 T α/2 (‖v1 − v2 ‖C ([0,T ]×[−h0 ,h0 ]) + ‖h′1 − h′2 ‖C ([0,T ]) + ‖g1′ − g2′ ‖C ([0,T ]) ) ≤
1 2
(‖v1 − v2 ‖C ([0,T ]×[−h0 ,h0 ]) + ‖h′1 − h′2 ‖C ([0,T ]) + ‖g1′ − g2′ ‖C ([0,T ]) ).
Thus, for this T , F is a contraction mapping. Now, by using theorem, we can make a the contraction mapping conclusion that there is a v(t , y), g (t ), h(t ) in ΓT such that F v(t , y), g (t ), h(t ) = v(t , y), g (t ), h(t ) . In other words,
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v(t , y), g (t ), h(t ) is the solution of problem (2.2) and therefore u(t , x), g (t ), h(t ) is the solution of problem (1.1). Moreover, by using the Schauder estimates, we have additional regularity of the solution, g (t ), h(t ) ∈ C 1+α/2 [0, T ] and u ∈ C 1+α/2,2+α ((0, T ] × (g (t ), h(t ))). Thus (u(t , x), g (t ), h(t )) is the classical solution of problem (1.1). Now we give the monotone behavior of the free boundary g (t ) and h(t ). Theorem 2.2. The free boundary g (t ) and h(t ) for problem (1.1) are strictly monotone decreasing and increasing, respectively; that is, for any solution in (0, T ], we have g ′ (t ) < 0,
h′ (t ) > 0,
for 0 < t ⩽ T .
Proof. Using the Hopf lemma on problem (1.1) yields that ux (t , g (t )) > 0,
ux (t , h(t )) < 0
for 0 < t ⩽ T .
Then, combining the two inequalities with the Stefan conditions gives the result.
Remark 2.1. If the initial function u0 is smooth and satisfies the consistency condition
−du′′0 (h0 ) − µu′0 (h0 )u′0 (h0 ) = up0 (0) and
− du′′0 (−h0 ) − µu′0 (−h0 )u′0 (−h0 ) = up0 (0),
then the solution (u, g , h) ∈ C 1+α/2,2+α ([0, T ] × [g (t ), h(t )]) × C 1+(1+α)/2 ([0, T ]) × C 1+(1+α)/2 ([0, T ]). Next, we present some fundamental results. Lemma 2.1 (The Maximum Principle). Let u(t , x) be a classical solution of the problem ut − d1u ≥ c (t , x)u(t , 0), u(t , x) = 0, x ∈ ∂ Q u(0, x) ≥ 0 x ∈ Q ,
(t , x) ∈ QT (2.17)
where QT = [0, T ] × Q , Q is a bounded domain in Rn , and 0 ∈ Q . If 0 ≤ c (t , x) ≤ c0 , then u(t , x) ≥ 0 for (t , x) ∈ Q T .
(2.18)
The proof is similar to that of the classical case; see Lemma 2.1 in [3]. ∗
Lemma 2.2 (The Comparison Principle). Suppose that T ∈ (0, ∞), h, g ∈ C 1 ([0, T ]), u ∈ C (DT ) ∩ C 1,2 (D∗T ), with D∗T = {(t , x) ∈ R2 : 0 < t ≤ T , g (t ) < x < h(t )} and
ut − duxx ≥ up (t , 0), t > 0, g (t ) < x < h(t ), ′ u = 0, h (t ) ≥ −µux , t > 0, x = h(t ), ′ u = 0, g (t ) ≤ −µux , t > 0, x = g (t ). If h0 ≤ h(0),
−h0 ≥ g (0) and u0 (x) ≤ u(0, x) in [−h0 , h0 ],
then the solution (u, g , h) of the free boundary problem (1.1) satisfies h(t ) ≤ h(t ),
g (t ) ≥ g (t ) in (0, T ],
u(t , x) ≤ u(t , x) for (t , x) ∈ (0, T ] × (g (t ), h(t )). Proof. Inspired by [18], we just need to make some obvious modification. For small ϵ > 0, we denote the unique solution of (1.1) with h0 replaced by h0 (1 − ϵ), and µ replaced by µ(1 − ϵ) by (uϵ , gϵ , hϵ ). We assert that hϵ (t ) < h(t ) for all t ∈ (0, T ]. Obviously this is true for small t > 0. If our assertion does not hold, then we can find a first t ∗ ≤ T such that hϵ (t ) < h(t ) for t ∈ (0, t ∗ ) and hϵ (t ∗ ) = h(t ∗ ). It follows that ′
h′ϵ (t ∗ ) ≥ h (t ∗ ). Now we make a comparison about uϵ and u over the region
Ωt ∗ := {(t , x) ∈ R2 : 0 < t ≤ t ∗ , 0 ≤ x < hϵ (t )}.
(2.19)
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The strong maximum principle implies that uϵ (t , x) < u(t , x) in Ωt ∗ . So w(t , x) := u(t , x) − uϵ (t , x) > 0 in Ωt ∗ ′ with w(t ∗ , hϵ (t ∗ )) = 0. It follows that wx (t ∗ , hϵ (t ∗ )) ≤ 0, from which we can obtain that h′ϵ (t ∗ ) < h (t ∗ ), in view of ∗ ∗ (uϵ )x (t , hϵ (t )) < 0. But this is a contradiction to (2.19). This shows our claim that hϵ (t ) < h(t ) for all t ∈ (0, T ]. Similarly, we can demonstrate that gϵ (t ) > g (t ) for all t ∈ (0, T ]. Now, we may apply the usual comparison principle over ΩT := {(t , x) ∈ R2 : 0 < t ≤ T , g (t ) < x < h(t )} to conclude that uϵ < u in ΩT . Since the unique solution of (1.1) depends continuously on the parameters in (1.1), as ϵ → 0, (uϵ , gϵ , hϵ ) converges to (u, g , h), the unique solution of (1.1). The desired result then holds by letting ϵ → 0 in the inequalities uϵ < u, gϵ > g and hϵ < h. 3. Finite time blowup In this section, we discuss the blowup behavior. First we present the following lemma. Lemma 3.1. A solution of problem (1.1) exists and is unique, and can be extended to [0, Tmax ), where Tmax ⩽ ∞. Furthermore, if Tmax < ∞, we have lim sup ‖u(t , x)‖L∞ ([0,t ]×[g (t ),h(t )]) = ∞.
(3.1)
t →Tmax
Proof. It is easy to obtain from the uniqueness and Zorn’s lemma that there exists a number Tmax such that [0, Tmax ) is the maximal time interval in which the solution exists. In order to complete the proof of our lemma, we have to demonstrate that, when Tmax < ∞, lim sup ‖u‖L∞ ([0,t ]×[g (t ),h(t )]) = ∞. t →Tmax
Next, we take advantage of the contradiction argument. Assume that Tmax < ∞ and that lim sup ‖u‖L∞ ([0,t ]×[g (t ),h(t )]) < ∞; t →Tmax
then there exist M1 , M2 > 0 such that Tmax ≤ M1 < ∞ and ‖u‖L∞ [g (t ),h(t )] ≤ M2 < ∞ for all t ∈ [0, Tmax ). In what follows, we first prove that h′ (t ) and g ′ (t ) are uniformly bounded in (0, Tmax ), i.e., h′ (t ) ≤ M3 and −g ′ (t ) ≤ M4 for all t ∈ (0, Tmax ), with some M3 , M4 independent of Tmax . To obtain h′ (t ) ≤ M3 , we define
Ω = ΩM := {(t , x) : 0 < t < Tmax , h(t ) − M −1 < x < h(t )} and construct an auxiliary function
w(t , x) := M2 [2M (h(t ) − x) − M 2 (h(t ) − x)2 ]. We will choose M so that w(t , x) ≥ u(t , x) holds over Ω . Direct computations yield that, for (t , x) ∈ Ω ,
wt = 2M2 Mh′ (t )(1 − M (h(t ) − x)) ≥ 0, p −wxx = 2M2 M 2 , up (t , 0) ≤ M2 . It follows that
wt − dwxx ≥ 2dM2 M 2 ≥ up (t , 0) in Ω if M 2 ≥
P −1
M2
2d
. On the other hand,
w(t , h(t ) − M −1 ) = M2 ≥ u(t , h(t ) − M −1 ),
w(t , h(t )) = 0 = u(t , h(t )).
Thus, if we can choose M such that u0 (x) ≤ w(0, x) for x ∈ [h0 − M −1 , h0 ], then we can apply the maximum principle to w − u over Ω to deduce that u(t , x) ≤ w(t , x) for (t , x) ∈ Ω . It would then follow that ux (t , h(t )) ≥ wx (t , h(t )) = −2MM2 ,
h′ (t ) = −µux (t , h(t )) ≤ M3 := 2MM2 µ.
Hence, now, we only have to find some M independent of Tmax such that u0 (x) ≤ w(0, x) for x ∈ [h0 − M −1 , h0 ]. We calculate
wx (0, x) = −2M2 M [1 − M (h0 − x)] ≤ −M2 M for x ∈ [h0 − (2M )−1 , h0 ]. 4‖u0 ‖ 1 C ([0,h0 ]) (M2 )P −1 , we will have wx (0, x) ≤ u′0 (x) for x ∈ [h0 − (2M )−1 , h0 ]. Therefore, upon choosing M := max , 2d 3M2 Since w(0, h0 ) = u0 (h0 ) = 0, the above inequality implies that w(0, x) ≥ u0 (x) for x ∈ [h0 − (2M )−1 , h0 ]. Moreover, for x ∈ [h0 − M −1 , h0 − (2M )−1 ], we have
w(0, x) ≥
3 4
M2 ,
u0 (x) ≤ ‖u0 ‖C 1 ([0,h0 ]) M −1 ≤
3 4
M2 .
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Therefore, u0 (x) ≤ w(0, x) for x ∈ [h0 −M −1 , h0 ], which tells us that h′ (t ) ≤ M3 in [0, Tmax ), with M3 = 2MM2 µ independent of Tmax . Similarly, we can obtain that −g ′ (t ) ≤ M4 in [0, Tmax ) and that M4 is also independent of Tmax . Since h′ (t ) and g ′ (t ) are uniformly bounded in [0, Tmax ), we can now find L∗ > 0 depending on M1 , M2 , M3 and M4 such that ‖u(t , x)‖C 1+α [g (t ),h(t )] ≤ L∗ for all t ∈ [0, Tmax ). Using Theorem 2.1 again, we can conclude that there exists a τ > 0 depending only on M3 , M4 and L∗ such that the solution of problem (1.1) with the initial time Tmax − τ2 can be extended uniquely to the time Tmax − τ2 + τ , which is a contradiction to the hypothesis. Thus the proof is ended. In what follows, we will give the blowup result of the solution of (1.1). Theorem 3.1. Let u(t , x) be a solution of problem (1.1). Then the solution u(t , x) blows up at a finite time provided that u0 (x) is sufficiently large in a neighborhood of 0. Proof. Here, we verify the above result mainly by the maximum principle and the comparison principle. Consider the following auxiliary problem:
vt − dvxx = v p (t , 0), −h0 < x < h0 , 0 < t < T , v(t , h0 ) = v(t , −h0 ) = 0, 0 < t < T , v(0, x) = u0 (x), 0 < t < T .
(3.2)
We assert that u(t , x) ⩾ v(t , x) on [−h0 , h0 ] × (0, T ] by using the comparison principle. On the other hand, according to Theorem 2.3 in [3], we know that under the assumption that u0 (x) is sufficiently large in a neighborhood of 0, v(t , x) will cease to exist at a finite time. Therefore, the blowup result also holds for u(t , x) as u(t , x) is an upper solution, which completes the proof. Although the above theorem provides a condition to ensure the finite-time blowup, the condition for the initial function u0 (x) is very rigid. Next we attempt to find some other conditions which are more specific about the size of u0 (x). Indeed, let w(x) be the solution of the problem
−dwxx = 1, −h0 < x < h0 , w(h0 ) = w(−h0 ) = 0.
(3.3) p
Then the nonnegative steady-state solutions of (3.2) can be written as v(x) = αw(x), where α = w(0) 1−p . Theorem 3.2. If u0 (x) > (
h20
/2d)
p 1−p
−1 2
h20
2d
2d
x +
, then the solution of (1.1) blows up in finite time.
1 2 Proof. Direct calculations yield that w(x) = − x + 2d Corollary 2.4 in [3].
h20 2d
p
and α = (h20 /2d) 1−p . Hence, the desired result follows from
The next blowup result will be used to derive the global slow solution. Theorem 3.3. Let ϕ(x) be the first eigenfunction of the eigenvalue problem
−ϕxx = λ1 ϕ, −h0 < x < h0 , (3.4) ϕ(h0 ) = ϕ(−h0 ) = 0, h with ϕ(x) > 0 in [−h0 , h0 ] and −0h ϕ(x)dx = 1. Then the solution of problem (1.1) with the initial function u0 (x) in the form 0 −1 1 h0 p−1 2 of M ϕ(x) ceases to exist in finite time provided that M > λ1 . −h0 ϕ (x)dx
Proof. Consider the following auxiliary problem:
vt − dvxx = f (v(t , 0)) := v p (t , 0), −h0 < x < h0 , 0 < t < T , v(t , h0 ) = v(t , −h0 ) = 0, 0 < t < T , v(0, x) = M ϕ(x), 0 < t < T .
(3.5)
It follows that u(t , x) ⩾ v(t , x) on [−h0 , h0 ] × (0, T ] from the comparison principle. Next, we prove that v(t , x) blows up at a finite time. Multiplying (3.5) by ϕ and then integrating over [−h0 , h0 ], we obtain F ′ (t ) + λ1 F (t ) =
∫
h0
−h0
f (v(t , 0))ϕ(x)dx,
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where F (t ) =
h0
∫
v(t , x)ϕ(x)dx.
−h0
Since f is convex and ϕ is positive within (−h0 , h0 ), we have
∫
h0
f (v)ϕ(x)dx ≥ f (F (t )).
−h0
At the same time, due to the concavity of ϕ(x) over [−h0 , h0 ], it is not difficult to deduce that v(t , 0) ≥ v(t , x) on [0, T ] × [−h0 , h0 ]. Hence, we have ∫ h0 f (v(t , x))ϕ(x)dx ≥ f (F (t )). F ′ (t ) + λ1 F (t ) ≥ −h0 1 p−1
h
As −0h u0 (x)ϕ(x)dx > λ1 0 is complete.
, we have f (F (0)) ≥ λ1 F (0), which implies that v(t , x) will blow up in a finite time. The proof
Remark 3.1. From Theorem 3.3, we know that, if u0 (x) is in the form of M ϕ(x), then the solution will grow to infinity provided that M is sufficiently large. 4. The global fast solution and the slow solution In this section, we mainly investigate the long-time behavior of the global solutions of (1.1). Here, we first give the existence of a fast solution by the following theorem. Theorem 4.1 (Fast Solution). Let u be a solution of problem (1.1). If u0 is small in the following sense:
‖ u0 ‖ ∞ ⩽
1 2
1
min [d/(16h20 )] p−1 , 1/(8µ) ,
then Tmax = ∞. Moreover, h∞ < ∞, g∞ > −∞, and there exist real numbers C , β > 0 depending on u0 such that
‖u(t )‖∞ ⩽ C e−β t ,
t ⩾ 0.
(4.1)
Proof. It suffices to the construct a suitable global supersolution. Motivated by [19], we define
ϑ(t ) = 2h0 (2 − e−γ t ),
λ(t ) = −ϑ(t ),
t ⩾ 0,
V (y) = 1 − y2 ,
−1 ⩽ y ⩽ 1,
and
v(x, t ) = ε e−β t V (x/ϑ),
λ(t ) ⩽ x ⩽ ϑ(t ), t ⩾ 0,
where γ , β and ε > 0 are to be determined later. Direct computation yields
vt − dvxx − v p (t , 0) = ε e−β t [−β V − xϑ ′ ϑ −2 V ′ − dϑ −2 V ′′ − ε p−1 e−β(p−1)t ] [ ] d p−1 ⩾ ε e−β t −β + − ε , 2 8h0
for all t > 0 and λ(t ) < x < ϑ(t ). On the other hand, we can easily get ϑ ′ (t ) = 2γ h0 e−γ t > 0 and −vx (t , ϑ(t )) = 2ε e−β t /ϑ(t ). Setting γ = β =
d 2
,
16h0 1 2 p−1 and ε ⩽ ε0 = min [d/(16h0 )] , 1/(8µ) , it follows that v − dvxx − v p (t , 0) ⩾ 0, λ(t ) < x < ϑ(t ), t > 0, t′ ϑ (t ) > −µvx (t , ϑ(t )), λ′ (t ) < −µvx (t , λ(t )), t > 0, v(t , ϑ(t )) = v(t , λ(t )) = 0, t > 0, ϑ(0) = 2h0 > h0 , λ(0) = −2h0 < −h0 . 1 Assuming that ‖u0 ‖∞ ⩽ 12 min [d/(16h20 )] p−1 , 1/(8µ) and choosing ε = 2‖u0 ‖∞ , we also get u0 (x) < v(x, 0) for −h0 ⩽ x ⩽ h0 . By applying the comparison principle, one can see that h(t ) < ϑ(t ), g (t ) > λ(t ) and that u(x, t ) < v(x, t ) for g (t ) ⩽ x ⩽ h(t ), as long as u exists. In particular, it follows from the continuation property (3.1) that u exists globally.
The proof is complete.
P. Zhou et al. / Nonlinear Analysis 74 (2011) 2523–2533
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Remark 4.1. The above result indicates that the free boundaries both converge to finite limits as t tends to infinity and that the solution u(t , x) decays uniformly to 0 at an exponential rate. The solution is therefore called a fast solution compared with the case in Theorem 4.2, where the double free boundary fronts will both grow up to infinity. Before giving the existence of a global slow solution, we need the following lemma, which provides a priori estimate for the global solution. Lemma 4.1. Let u be a solution of problem (1.1) with u0 (x) in the form of δϕ(x). If Tmax = ∞, then there exists a constant C = C (‖u0 ‖C 1+α , h0 , 1/h0 ), such that sup ‖u(t , x)‖L∞ (g (t ),h(t )) ⩽ C , t ⩾0
where C remains bounded for ‖u0 ‖C 1+α , h0 , and 1/h0 bounded. Proof. First, from the local theory for problem (1.1), for each M > 1, there exists σ > 0 such that, if ‖u0 ‖C 1+α < M and 1/M < h0 < M, then ‖u(t , x)‖L∞ < 2M on [0, σ ]. In what follows, we use the contradiction argument. If the result is false, then there exists an M > 0 and a sequence of global solutions (un , gn , hn ) of (1.1), such that 1/M < gn (0) < M ,
‖un (0, x)‖C 1+α [−h0 ,h0 ] < M
and sup ‖un (t , x)‖L∞ (gn (t ),hn (t )) → ∞ as n → ∞. t ⩾0
For all large n, there exist tn ⩾ σ and xn ∈ (gn (t ), hn (t )) such that 3 4
sup ‖un (t , x)‖L∞ (gn (t ),hn (t )) = un (tn , xn ). t ⩾0
We claim that xn = 0. In fact, we set v(t , x) = ux (t , x). Since now u0 = λϕ is symmetric and concave on [−h0 , h0 ], we have h(t ) = −g (t ), and, for any t ∈ [0, ∞), the corresponding solution is also symmetric over (g (t ), h(t )). Hence, we can deduce that v(t , 0) = ux (t , 0) = 0. Then v(t , x) satisfies
v − dvxx = 0, t > 0, 0 < x < h(t ), t v(t , h(t )) = ux (t , h(t )) ≤ 0, t > 0, v(t , 0) = ux′ (t , 0) = 0, t > 0, v(0, x) = u0 (x) ≤ 0, 0 ≤ x ≤ h0 .
(4.2)
By the maximum principle, we can deduce that v(t , x) ≤ 0 for x ∈ (0, h(t )). Similarly, we can obtain that v(t , x) ≥ 0 for x ∈ (−g (t ), 0). Therefore, u(t , x) ≤ u(t , 0) over [0, +∞) × [g (t ), h(t )]. Thus, we have xn = 0, which implies that sup ‖un (t , x)‖L∞ (gn (t ),hn (t )) = un (tn , 0) , ϱn . t ⩾0
−(p−1)/2
We define λn = ϱn ; then it is evident to see that λn → 0 as n → ∞. We extend un (t , ·) by 0 on (hn (t ), ∞) and (−∞, gn (t )) and define the rescaled function
vn (τ , y) = λ2n un (tn + λ2n τ , λn y)
(4.3)
2 for (τ , y) ∈ Dn = {(τ , y) : −λ− n tn ⩽ τ ⩽ 0 and − ∞ < y < ∞}. Also, we denote 1 2 y 1 = λ− n g (tn + λn τ ),
1 2 y2 (τ ) = λ− n h(tn + λn τ )
and 2 Dn = {(τ , y) : −λ− n tn ⩽ τ ⩽ 0 and y1 (τ ) ⩽ y < y2 (τ )},
which corresponds to the domain {g (t ) < x < h(t )}. The function vn satisfies vn (0, 0) = 1, 0 ⩽ vn ⩽
∂τ vn − d∂y2 vn = vnp (τ , 0),
4 3
and
(y, τ ) ∈ Dn .
(4.4) p Lloc
Similarly to Lemma 2.1 in [1], there exists a subsequence {vnk } of {vn } such that {vnk } converges in ((−∞, 0] × (−∞, +∞)) to a function w(τ , y) ∈ Lploc ((−∞, 0] × (−∞, +∞)), and {vnk (0, y)} converges in Cloc (−∞, +∞) to a function z (y) ∈ C (−∞, +∞), which satisfies z (0) = 1. Moreover, similarly to Lemmas 2.2 and 2.3 in [1], wt = 0 in D′ ((−∞, 0) × (−∞, +∞)), and there is a function w(y) ⩾ 0 which is bounded, continuous on (−∞, ∞), and satisfies that −wyy = wp (0); hence w is concave. Therefore, w ≡ 0, which leads to a contradiction to the fact that w(0) = z (0) = 1. This completes the proof of Lemma 4.1.
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The above result indicates that the global solution is uniformly bounded while the following lemma will point out that the global solution decays uniformly to 0. Lemma 4.2. Under the same condition as that in Lemma 4.1, the solution satisfies lim ‖u(t , x)‖L∞ (g (t ),h(t )) = 0.
t →+∞
Proof. Assume that k := lim supt →+∞ ‖u(t , x)‖L∞ (g (t ),h(t )) > 0 by contradiction. It follows from Lemma 4.1 that k < +∞. Let t0 > 0 be such that sup ‖u(t , x)‖L∞ (g (t ),h(t )) ⩽
[t0 ,+∞)
3 2
k.
Then there exists a sequence tn → +∞ such that ‖u(tn , x)‖L∞ (g (tn ),h(tn )) ⩾
3 k. 4
As in Lemma 4.1, we can obtain
‖u(tn , x)‖L∞ (g (tn ),h(tn )) = u(tn , 0) and then define λn = u−(p−1)/2 (tn , xn ). We extend u(t , ·) by 0 on (−∞, g (t )) and (h(t ), ∞) and define the rescaled function 2
vn (τ , y) = λnp−1 u(tn + λ2n τ , λn y)
(4.5)
2 for (τ , y) ∈ Dn = {(τ , y) : λ− n (t0 − tn ) ⩽ τ ⩽ 0 and − ∞ < y < ∞}. Also, we denote 2 1 y1 (τ ) = λ− n g (tn + λn ),
1 2 y2 (τ ) = λ− n h(tn + λn τ )
and 2 Dn = {(τ , y) : λ− n (t0 − tn ) ⩽ τ ⩽ 0 and y1 ⩽ y < y2 (τ )}.
Therefore, the function vn satisfies vn (0, 0) = 1, 0 ⩽ vn ⩽ 2 ∈ Dn , and
∂τ vn − d∂y2 vn = vnp (τ , 0),
(τ , y) ∈ Dn .
(4.6)
Similarly to Lemmas 2.1–2.3 in [1], we have obtained a function w(y) ⩾ 0, bounded and continuous on (−∞, ∞), which satisfies that −wyy = w p (0). Therefore, w ≡ 0, which contradicts the fact that w(0) = 1 since vn (0, 0) = 1. The proof is ended. Theorem 4.2 (Slow Solution). Let ϕ(x) be the first eigenfunction of the eigenvalue problem (3.4). Then there exists λ > 0 such that the solution of (1.1) with initial data u0 = λϕ is a global slow solution, which satisfies that h∞ = −g∞ = ∞. Proof. In order to emphasize the dependence of u on the initial data when necessary, we denote the solution of (1.1) by u(u0 ; ·). So do g (t ), h(t ), g∞ , h∞ and the maximal existence time T . We first claim that when u0 = λϕ we have h(t ) = −g (t ), since the initial function is symmetric on [−h0 , h0 ], which implies that h(t ) and g (t ) are both finite or infinite at the same time. Motivated by [1], we define
Σ = {λ > 0 : T (λϕ) = ∞ and h∞ (λϕ) < ∞}. According to Theorem 4.1, we know that λ ∈ Σ if λ is small, so Σ is not empty. In contrast, when λ is sufficiently large, it follows from Theorem 3.3 that the corresponding solution will blow up, i.e., T (λϕ) < ∞; hence Σ is bounded. Let
λ∗ = sup Σ ∈ (0, ∞),
v = u(λ∗ ϕ; ·),
σ = h(λ∗ ϕ; ·) and τ = T (λ∗ ϕ).
First, we claim that τ = ∞. In fact, by continuous dependence (see [20,21]), for any fixed t ∈ [0, τ ), u(λϕ; t , x) converges to v(t , x) in L∞ (−∞, ∞) and h(λϕ; t ) → σ (t ) as λ ↑ λ∗ . Here we extend u(t , x) by 0 for x ∈ (−∞, g (t )) (h(t ), ∞). It follows from Lemma 4.1 that ‖v(t )‖∞ ≤ C for all t ∈ [0, τ ), because T (λϕ) = ∞ for all λ ∈ (0, λ∗ ). Thus we have τ = ∞, since nonglobal solutions should satisfy lim supt →T ‖u(t , x)‖L∞ (g (t ),h(t )) = ∞. Next we claim that σ∞ = ∞. In what follows, we use a contradiction argument. Assume thatσ∞ < ∞. Since ‖v(t )‖∞
→ 0 as t → ∞ by Lemma 4.2, we can choose t0 sufficiently large such that ‖v(t0 )∞ ‖ <
1 2
1
min [d/(16h20 )] p−1 , 1/(8µ) .
By continuous dependence, we can deduce that
‖u(λφ; t0 , x)‖L∞ (g (t0 ),h(t0 )) ⩽
1 2
1
min [d/(16h20 )] p−1 , 1/(8µ)
for λ > λ∗ sufficiently close to λ∗ . But this implies that h∞ (λφ) < ∞ by Theorem 4.1, which is a contradiction to the definition of λ∗ . The proof is complete.
P. Zhou et al. / Nonlinear Analysis 74 (2011) 2523–2533
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References [1] M. Fila, P. Souplet, Existence of global solutions with slow decay and unbounded free boundary for a superlinear Stefan problem, Interfaces Free Bound. 3 (2001) 337–344. [2] H. Ghidouche, P. Souplet, D. Tarzia, Decay of global solutions, stability and blow-up for a reaction–diffusion problem with free boundary, Proc. Amer. Math. Soc. 129 (2001) 781–792. [3] J.M. Chadam, A. Peirce, H.M. Yin, The blowup property of solutions to some diffusion equations with localized nonlinear reactions, J. Math. Anal. Appl. 169 (1992) 313–328. [4] K. Bimpong-Bota, P. Ortoleva, J. Ross, Far-from-equilibrium phenomena at local sites of reaction, J. Chem. Phys. 60 (1974) 3124–3133. [5] P. Ortoleva, J. Ross, Local structures in chemical reactions with heterogeneous catalysis, J. Chem. Phys. 56 (1972) 4397. [6] J. Crank, Free and Moving Boundary Problem, Clarendon Press, Oxford, 1984. [7] J. Goodman, D.N. Ostrov, On the early exercise boundary of the American put option, SIAM J. Appl. Math. 62 (2002) 1823–1835. [8] L.S. Jiang, B.J. Bian, F.H. Yi, A parabolic variational inequality arising from the valuation of fixed rate mortgages, European J. Appl. Math. 16 (2005) 361–383. [9] X.F. Chen, A. Friedman, A free boundary problem arising in a model of wound healing, SIAM J. Math. Anal. 32 (2000) 778–800. [10] V. Guyonne, L. Lorenzi, Instability in a flame ball problem, Discrete Contin. Dyn. Syst. Ser. B 7 (2007) 315–350. [11] A. Friedman, Free boundary problems associated with multiscale tumor models, Math. Model. Nat. Phenom. 4 (2009) 134–155. [12] D. Hilhorst, M. Mimura, R. Schatzle, Vanishing latent heat limit in a Stefan-like problem arising in biology, Nonlinear Anal. RWA 4 (2003) 261–285. [13] K.I. Kim, Z.G. Lin, A free boundary problem for a parabolic system describing an ecological model, Nonlinear Anal. RWA 10 (2009) 428–436. [14] K.I. Kim, Z.G. Lin, Z. Ling, Global existence and blowup of solutions to a free boundary problem for mutualistic model, Sci. China Math. 53 (2010) 2085–2095. [15] Z.G. Lin, A free boundary problem for a predator–prey model, Nonlinearity 20 (2007) 1883–1892. [16] M. Mimura, Y. Yamada, S. Yotsutani, Free boundary problems for some reaction–diffusion equations, Hiroshima Math. J. 17 (1987) 241–280. [17] O.A. Ladyzenskaja, V.A. Solonnikov, N.N. Ural’ceva, Linear and Quasilinear Equations of Parabolic Type, Amer. Math. Soc., Providence, RI, 1968. [18] Y.H. Du, Z.G. Lin, Spreading-vanishing dichotomy in the diffusive logistic model with a free boundary, SIAM J. Math. Anal. 42 (2010) 377–405. [19] R. Ricci, D.A. Tarzia, Asymptotic behavior of the solutions of the dead-core problem, Nonlinear Anal. 13 (1989) 405–411. [20] A. Fasano, M. Primicerio, Free boundary problems for nonlinear parabolic equations with nonlinear free boundary conditions, J. Math. Anal. Appl. 72 (1979) 247–273. [21] P. Souplet, Stability and continuous dependence of solutions to one-phase Stefan problems for semilinear parabolic equations, Port. Math. 59 (2002) 315–323.
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Nonlinear Analysis journal homepage: www.elsevier.com/locate/na
Global existence and blowup of a localized problem with free boundary✩ Peng Zhou, Jie Bao, Zhigui Lin ∗ School of Mathematical Science, Yangzhou University, Yangzhou 225002, PR China
article
abstract
info
Article history: Received 15 July 2010 Accepted 27 November 2010
This paper is concerned with a double fronts free boundary problem for the heat equation with a localized nonlinear reaction term. The local existence and uniqueness of the solution are given by applying the contraction mapping theorem. Then we present some conditions so that the solution blows up in finite time. Finally, the long-time behavior of the global solution is discussed. We show that the solution is global and fast if the initial data is small and that a global slow solution is possible when the initial data is suitably large. © 2010 Elsevier Ltd. All rights reserved.
MSC: 35K20 35R35 92B05 Keywords: Free boundary Blowup Global fast solution Global slow solution Localized
1. Introduction In this paper, we investigate the behavior of the positive solution u(t , x) of the following localized problem with double fronts free boundary
u − duxx = up (t , 0), t > 0, g (t ) < x < h(t ), t u(t , g (t )) = 0, g ′ (t ) = −µux (t , g (t )), t > 0, h′ (t ) = −µux (t , h(t )), t > 0, u(t , h(t )) = 0, g (0) = −h0 , h(0) = h0 , u(0, x) = u0 (x), −h0 ≤ x ≤ h0 ,
(1.1)
where both x = g (t ) and x = h(t ) are moving boundaries to be determined, h0 > 0, P > 1, d and µ are positive constants, and the initial function u0 satisfies
u0 ∈ C 2 ([−h0 , h0 ]), u0 (−h0 ) = u0 (h0 ) = 0,
and u0 > 0 in (−h0 , h0 ).
(1.2)
In [1,2], the authors considered a Stefan problem with nonlocal superlinear term, and exhibited an energy condition under which the solution would blow up in finite time. In addition, they proved that all global solutions are bounded and decay uniformly to 0. Moreover, they showed that the free boundary converges to a finite limit and the solution decays at an exponential rate, or the free boundary grows up to infinity and the decay rate is at most polynomial. For the localized equation ut − 1u = f (u(t , x0 )) ✩ The work is partially supported by PRC grant NSFC 11071209, 10801115 and also by NSF 10KJB110011 of Jiangsu Province.
∗
Corresponding author. E-mail address: [email protected] (Z. Lin).
0362-546X/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2010.11.047
(1.3)
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P. Zhou et al. / Nonlinear Analysis 74 (2011) 2523–2533
in a fixed domain, much work has been done to consider the long-time behavior of its solution. For example, the authors in [3] investigated this equation with Dirichlet and Neumann boundary conditions, respectively, as well as the corresponding Cauchy problem. They showed that under certain conditions the solution would cease to exist in finite time. Moreover, they proved that the set of all blowup points is the whole region, and estimated the growth rate of solutions near the blowup time. As we know, Eq. (1.3) describes some physical phenomena in which the nonlinear reaction in a dynamical system takes place only at a single (or sometimes several) site(s). As an example, the influence of defect structures on a catalytic surface can be modeled by a similar equation. One can find more detailed information about the physical derivation by consulting [4,5]. Recently, the free boundary problem has attracted much attention in many areas, for instance, the decrease of oxygen in a muscle in the vicinity of a clotted blood vessel [6], the American option pricing problem [7,8], wound healing [9], combustion under gravity conditions [10], tumor growth [11] and the dynamics of a population [12–16]. In this paper, we consider the localized problem with free boundary (1.1) and pay much attention to the blowup property of the solution and the long-time behavior of the global solutions. We will give some sufficient conditions ensuring the existence of a fast solution and a slow solution. Here, we say that the solution exists globally if Tmax = +∞, whereas if the solution ceases to exist for some finite time, that is, Tmax < +∞ and limt →Tmax ‖u(t , x)‖L∞ ([0,t ]×[g (t ),h(t )]) = +∞, we say that the solution blows up. If Tmax = ∞, limt →∞ h(t ) < ∞ and limt →∞ g (t ) < ∞, then the solution is called fast solution, since that the solution decays uniformly to 0 at an exponential rate, while if Tmax = ∞, limt →∞ h(t ) = ∞ and limt →∞ g (t ) = ∞, it is called slow solution, whose decay rate is at most polynomial; see [1,2] for details. The paper is organized as follows. First, local existence and uniqueness are exhibited in Section 2 as well as some preliminary results such as the maximum principle. Section 3 is devoted to the blowup property. Our result shows that the solution will blow up if the initial data is sufficiently large. Section 4 deals with the long-time behavior of global solutions, including the existence of a global fast solution and a global slow solution. 2. Local existence and uniqueness In this section, by using the contraction mapping theorem, we first give the following local existence and uniqueness result. And then we use the Hopf lemma to verify the monotonicity of the double free boundary fronts. Lastly, some preliminary results about the comparison principle are presented. Theorem 2.1. For any given u0 satisfying (1.2) and any α ∈ (0, 1), there is a T > 0 such that problem (1.1) admits a unique solution
(u, h, g ) ∈ C (1+α)/2,1+α (DT ) × C 1+α/2 ([0, T ]) × C 1+α/2 ([0, T ]); moreover,
‖u‖C (1+α)/2,1+α (DT ) + ‖h‖C 1+α/2 ([0,T ]) + ‖g ‖C 1+α/2 ([0,T ]) ≤ C ,
(2.1)
where DT = {(t , x) ∈ R2 : t ∈ [0, T ], x ∈ [g (t ), h(t )]}, C and T depend only on h0 , α and ‖u0 ‖C 2 ([−h0 ,h0 ]) . Proof. We first make a change of variable to straighten the free boundary. Let y=
2h0 x h(t ) − g (t )
−
h0 (h(t ) + g (t )) h(t ) − g (t )
,
v(t , y) = u(t , x).
Then problem (1.1) becomes
h(t ) + g (t ) p , t > 0, − h0 < y < h0 , v t , −h 0 t − Avy − Bvyy = v h(t ) − g (t ) 2h0 µ ∂v v = 0, h′ (t ) = − , t > 0, y = h0 , h(t ) − g (t ) ∂ y 2h0 µ ∂v v = 0, g ′ (t ) = − , t > 0, y = −h0 , h(t ) − g (t ) ∂ y h(0) = h0 , g (0) = −h0 , v(0, y) = v0 (y) := u0 (y), −h0 ≤ y ≤ h0 , h′ (t )−g ′ (t )
4h2 d
h′ (t )+g ′ (t )
(2.2)
where A = A(h, g , y) = y h(t )−g (t ) + h0 h(t )−g (t ) and B = B(h, g ) = (h(t )−0g (t ))2 . This transformation changes the free boundaries x = h(t ) and x = g (t ) to the fixed lines x = h0 and x = −h0 , respectively, but at the expense of making the equation more complex, as the coefficients in the first equation of (2.2) now include the unknown function h(t ) and g (t ). We denote k1 = −µv0′ (−h0 ) and k2 = −µv0′ (h0 ), and set
GT = g ∈ C 1 [0, T ] : g (0) = −h0 , g ′ (0) = k1 , ‖g ′ (t ) − k1 ‖ ⩽ 1 ,
HT = h ∈ C 1 [0, T ] : h(0) = h0 , h′ (0) = k2 , ‖h′ (t ) − k2 ‖ ⩽ 1 ,
P. Zhou et al. / Nonlinear Analysis 74 (2011) 2523–2533
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UT = v ∈ C ([0, T ] × [−h0 , h0 ]) : v(0, y) = v0 (y), ‖v − v0 ‖C ([0,T ]×[−h0 ,h0 ]) ⩽ 1 . It is not difficult to see that ΓT := UT × GT × HT is a complete metric space with the metric
D ((v1 , g1 , h1 ), (v2 , g2 , h2 )) = ‖v1 − v2 ‖C ([0,T ]×[−h0 ,h0 ]) + ‖g1′ − g2′ ‖C ([0,T ]) + ‖h′1 − h′2 ‖C ([0,T ]) . Let us notice that, for g1 , g2 ∈ GT , since g1 (0) = g2 (0) = −h0 , we have
‖g1 − g2 ‖C ([0,T ]) ≤ T ‖g1′ − g2′ ‖C ([0,T ]) .
(2.3)
Similarly we have that, for h1 , h2 ∈ HT ,
‖h1 − h2 ‖C ([0,T ]) ≤ T ‖h′1 − h′2 ‖C ([0,T ]) .
(2.4)
Next, by using standard L theory and the Sobolev imbedding theorem [17], we can obtain, for any (v, g , h) ∈ ΓT , the following initial boundary value problem: p
h+g p , −h 0 < y < h 0 , 0 < t < T , v˜ t − Av˜ y − Bv˜ yy = v t , −h0 h−g v˜ (t , h0 ) = v˜ (t , −h0 ) = 0, 0 < t < T , v˜ (0, y) = v0 (y) ⩾ 0, −h0 ⩽ y ⩽ h0
(2.5)
admits a unique solution v˜ ∈ C (1+α)/2,1+α ([0, T ] × [−h0 , h0 ]) and
‖˜v ‖C (1+α)/2,1+α ([0,T ]×[−h0 ,h0 ]) ⩽ C ‖˜v ‖W 1,2,p ([0,T ]×[−h0 ,h0 ]) ⩽ C1 ,
(2.6)
3 where A, B is defined as that in (2.2), p = 1−α , and C1 is a constant depending on α, h0 and ‖u0 ‖C 1+α [−h0 ,h0 ] . Now, we define s˜(t ) as the following:
s˜ = (2h0 ) − 4h0 µ 2
2
t
∫
v˜ y (τ , h0 ) − v˜ y (τ , −h0 )dτ ,
(2.7)
0
and we then have s˜(0) = 2h0 . Based on s˜(t ), we now define h˜ (t ) − h0 =
t
∫ 0
−2h0 µ v˜ y (τ , h0 )dτ s˜(τ )
(2.8)
−2h0 µ v˜ y (τ , −h0 )dτ . s˜(τ )
(2.9)
and g˜ (t ) + h0 =
t
∫ 0
According to (2.8) and (2.9), we can easily get h˜′ (t ) =
−2h0 µ v˜ y (t , h0 ) s˜(t )
g˜′ (t ) =
−2h0 µ v˜ y (t , −h0 ), s˜(t )
and
which also imply
(h˜ ′ (t ) − g˜ ′ (t ))˜s(t ) = s˜(t )˜s′ (t ) due to (2.7). Thus we have h˜ ′ (t ) − g˜ ′ (t ) = s˜′ (t ). Since h˜ (0) − g˜ (0) = s˜(0) = 2h0 , we get h˜ (t ) − g˜ (t ) = s˜(t ). Now, we obtain the following two equations: h˜′ (t ) =
−2h0 µ v˜ (t , h0 ) ˜h(t ) − g˜ (t ) y
(2.10)
g˜′ (t ) =
−2h0 µ v˜ (t , −h0 ), ˜h(t ) − g˜ (t ) y
(2.11)
and
with h˜ (0) = −˜g (0) = h0 , h˜ ′ (0) = k2 , and g˜ ′ (0) = k1 , and hence h˜ ′ (t ), g˜ ′ (t ) ∈ C α/2 ([0, T ]), with
‖h˜ ′ (t )‖C α/2 ([0,T ]) ⩽ C2 := µC1
(2.12)
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and
‖˜g ′ (t )‖C α/2 ([0,T ]) ⩽ C2 := µC1 .
(2.13)
Next we define a map F : ΓT −→ C ([0, T ] × [−h0 , h0 ]) × C 1 [0, T ] × C 1 [0, T ] by
F (v(t , y), g (t ), h(t )) = (˜v (t , y), g˜ (t ), h˜ (t )). It is obvious that (v(t , y), g (t ), h(t )) ∈ ΓT is a fixed point of F if and only if it solves (2.2). By (2.6), (2.12) and (2.13), we have
‖h˜ ′ − k2 ‖C ([0,T ]) ≤ ‖h˜ ′ ‖C α/2 ([0,T ]) T α/2 ≤ µC1 T α/2 , ‖˜g ′ − k1 ‖C ([0,T ]) ≤ ‖˜g ′ ‖C α/2 ([0,T ]) T α/2 ≤ µC1 T α/2 , ‖˜v − u0 ‖C ([0,T ]×[−h0 ,h0 ]) ≤ ‖˜v − u0 ‖C (1+α)/2,0 ([0,T ]×[−h0 ,h0 ]) T (1+α)/2 ≤ C1 T (1+α)/2 . −2/(1+α)
Therefore, if we take T ≤ min{(µC1 )−2/α , C1 }, then F maps ΓT into itself. To demonstrate that F is a contraction mapping over ΓT for T > 0 sufficiently small, we take (vi , gi , hi ) ∈ ΓT (i = 1, 2), and denote (v i , g i , hi ) = F (vi , gi , hi ). Then, it follows from (2.6), (2.12) and (2.13) that
‖v i ‖C (1+α)/2,1+α ([0,T ]×[−h0 ,h0 ]) ⩽ C ‖v i ‖W 1,2,p ([0,T ]×[−h0 ,h0 ]) ⩽ C1 , ‖g ′i (t )‖C α/2 [0,T ] ⩽ C2 ,
′
‖hi (t )‖C α/2 [0,T ] ⩽ C2 .
By setting W = v 1 − v 2 and performing some direct computation, we can deduce that W (t , y) satisfies Wt − A(h1 , g1 , y)Wy − B(h1 , g1 )Wyy = [A(h1 , g1 , y) − A(h2 , g2 , y)]v 2,y + [B(h1 , g1 ) − B(h2 , g2 )]v 2,yy h1 + g1 h2 + g2 + v1p t , −h0 − v2p t , −h0 , t > 0, h0 < y < h0 , h1 − g1 h2 − g2 W (t , −h0 ) = 0, W (0, y) = 0,
W ( t , h0 ) = 0 ,
t > 0,
−h 0 ≤ y ≤ h 0 .
p
Applying the L estimates for parabolic equations and Sobolev’s imbedding theorem, we obtain
‖v 1 − v 2 ‖C (1+α)/2,1+α ([0,T ]×[−h0 ,h0 ]) ≤ C3 (‖v1 − v2 ‖C ([0,T ]×[−h0 ,h0 ]) + ‖h1 − h2 ‖C 1 ([0,T ]) + ‖g1 − g2 ‖C 1 ([0,T ]) ),
(2.14)
where C3 depends on C1 and C2 , and the functions A, B and C are defined in (2.2). Taking the difference of the equations for h1 , h2 , g 1 and g 2 results in
′ ′ ‖h1 − h2 ‖C α/2 ([0,T ]) ≤ µ ‖w1,y − w2,y ‖C α/2,0 ([0,T ]×[−h0 ,h0 ])
(2.15)
‖g ′1 − g ′2 ‖C α/2 ([0,T ]) ≤ µ ‖w1,y − w2,y ‖C α/2,0 ([0,T ]×[−h0 ,h0 ]) .
(2.16)
and
Combining inequalities (2.14)–(2.16), and assuming that T ≤ 1, we get ′
′
‖v 1 − v 2 ‖C (1+α)/2,1+α ([0,T ]×[−h0 ,h0 ]) + ‖h1 − h2 ‖C α/2 ([0,T ]) + ‖g ′1 − g ′2 ‖C α/2 ([0,T ]) ≤ C4 (‖v1 − v2 ‖C ([0,T ]×[−h0 ,h0 ]) + ‖h′1 − h′2 ‖C [0,T ] + ‖g1′ − g2′ ‖C [0,T ] ) with C4 depending on C3 and µ. Hence, for
2/α 1 −2/(1+α) −2/α T := min 1, , (µC1 ) , C1 , 2C4
we have ′
′
‖v 1 − v 2 ‖C ([0,T ]×[−h0 ,h0 ]) + ‖h1 − h2 ‖C ([0,T ]) + ‖g ′1 − g ′2 ‖C ([0,T ]) ′
′
≤ T (1+α)/2 ‖v 1 − v 2 ‖C (1+α)/2,1+α ([0,T ]×[−h0 ,h0 ]) + T α/2 (‖h1 − h2 ‖C α/2 ([0,T ]) + ‖g ′1 − g ′2 ‖C α/2 ([0,T ]) ) ≤ C4 T α/2 (‖v1 − v2 ‖C ([0,T ]×[−h0 ,h0 ]) + ‖h′1 − h′2 ‖C ([0,T ]) + ‖g1′ − g2′ ‖C ([0,T ]) ) ≤
1 2
(‖v1 − v2 ‖C ([0,T ]×[−h0 ,h0 ]) + ‖h′1 − h′2 ‖C ([0,T ]) + ‖g1′ − g2′ ‖C ([0,T ]) ).
Thus, for this T , F is a contraction mapping. Now, by using theorem, we can make a the contraction mapping conclusion that there is a v(t , y), g (t ), h(t ) in ΓT such that F v(t , y), g (t ), h(t ) = v(t , y), g (t ), h(t ) . In other words,
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v(t , y), g (t ), h(t ) is the solution of problem (2.2) and therefore u(t , x), g (t ), h(t ) is the solution of problem (1.1). Moreover, by using the Schauder estimates, we have additional regularity of the solution, g (t ), h(t ) ∈ C 1+α/2 [0, T ] and u ∈ C 1+α/2,2+α ((0, T ] × (g (t ), h(t ))). Thus (u(t , x), g (t ), h(t )) is the classical solution of problem (1.1). Now we give the monotone behavior of the free boundary g (t ) and h(t ). Theorem 2.2. The free boundary g (t ) and h(t ) for problem (1.1) are strictly monotone decreasing and increasing, respectively; that is, for any solution in (0, T ], we have g ′ (t ) < 0,
h′ (t ) > 0,
for 0 < t ⩽ T .
Proof. Using the Hopf lemma on problem (1.1) yields that ux (t , g (t )) > 0,
ux (t , h(t )) < 0
for 0 < t ⩽ T .
Then, combining the two inequalities with the Stefan conditions gives the result.
Remark 2.1. If the initial function u0 is smooth and satisfies the consistency condition
−du′′0 (h0 ) − µu′0 (h0 )u′0 (h0 ) = up0 (0) and
− du′′0 (−h0 ) − µu′0 (−h0 )u′0 (−h0 ) = up0 (0),
then the solution (u, g , h) ∈ C 1+α/2,2+α ([0, T ] × [g (t ), h(t )]) × C 1+(1+α)/2 ([0, T ]) × C 1+(1+α)/2 ([0, T ]). Next, we present some fundamental results. Lemma 2.1 (The Maximum Principle). Let u(t , x) be a classical solution of the problem ut − d1u ≥ c (t , x)u(t , 0), u(t , x) = 0, x ∈ ∂ Q u(0, x) ≥ 0 x ∈ Q ,
(t , x) ∈ QT (2.17)
where QT = [0, T ] × Q , Q is a bounded domain in Rn , and 0 ∈ Q . If 0 ≤ c (t , x) ≤ c0 , then u(t , x) ≥ 0 for (t , x) ∈ Q T .
(2.18)
The proof is similar to that of the classical case; see Lemma 2.1 in [3]. ∗
Lemma 2.2 (The Comparison Principle). Suppose that T ∈ (0, ∞), h, g ∈ C 1 ([0, T ]), u ∈ C (DT ) ∩ C 1,2 (D∗T ), with D∗T = {(t , x) ∈ R2 : 0 < t ≤ T , g (t ) < x < h(t )} and
ut − duxx ≥ up (t , 0), t > 0, g (t ) < x < h(t ), ′ u = 0, h (t ) ≥ −µux , t > 0, x = h(t ), ′ u = 0, g (t ) ≤ −µux , t > 0, x = g (t ). If h0 ≤ h(0),
−h0 ≥ g (0) and u0 (x) ≤ u(0, x) in [−h0 , h0 ],
then the solution (u, g , h) of the free boundary problem (1.1) satisfies h(t ) ≤ h(t ),
g (t ) ≥ g (t ) in (0, T ],
u(t , x) ≤ u(t , x) for (t , x) ∈ (0, T ] × (g (t ), h(t )). Proof. Inspired by [18], we just need to make some obvious modification. For small ϵ > 0, we denote the unique solution of (1.1) with h0 replaced by h0 (1 − ϵ), and µ replaced by µ(1 − ϵ) by (uϵ , gϵ , hϵ ). We assert that hϵ (t ) < h(t ) for all t ∈ (0, T ]. Obviously this is true for small t > 0. If our assertion does not hold, then we can find a first t ∗ ≤ T such that hϵ (t ) < h(t ) for t ∈ (0, t ∗ ) and hϵ (t ∗ ) = h(t ∗ ). It follows that ′
h′ϵ (t ∗ ) ≥ h (t ∗ ). Now we make a comparison about uϵ and u over the region
Ωt ∗ := {(t , x) ∈ R2 : 0 < t ≤ t ∗ , 0 ≤ x < hϵ (t )}.
(2.19)
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The strong maximum principle implies that uϵ (t , x) < u(t , x) in Ωt ∗ . So w(t , x) := u(t , x) − uϵ (t , x) > 0 in Ωt ∗ ′ with w(t ∗ , hϵ (t ∗ )) = 0. It follows that wx (t ∗ , hϵ (t ∗ )) ≤ 0, from which we can obtain that h′ϵ (t ∗ ) < h (t ∗ ), in view of ∗ ∗ (uϵ )x (t , hϵ (t )) < 0. But this is a contradiction to (2.19). This shows our claim that hϵ (t ) < h(t ) for all t ∈ (0, T ]. Similarly, we can demonstrate that gϵ (t ) > g (t ) for all t ∈ (0, T ]. Now, we may apply the usual comparison principle over ΩT := {(t , x) ∈ R2 : 0 < t ≤ T , g (t ) < x < h(t )} to conclude that uϵ < u in ΩT . Since the unique solution of (1.1) depends continuously on the parameters in (1.1), as ϵ → 0, (uϵ , gϵ , hϵ ) converges to (u, g , h), the unique solution of (1.1). The desired result then holds by letting ϵ → 0 in the inequalities uϵ < u, gϵ > g and hϵ < h. 3. Finite time blowup In this section, we discuss the blowup behavior. First we present the following lemma. Lemma 3.1. A solution of problem (1.1) exists and is unique, and can be extended to [0, Tmax ), where Tmax ⩽ ∞. Furthermore, if Tmax < ∞, we have lim sup ‖u(t , x)‖L∞ ([0,t ]×[g (t ),h(t )]) = ∞.
(3.1)
t →Tmax
Proof. It is easy to obtain from the uniqueness and Zorn’s lemma that there exists a number Tmax such that [0, Tmax ) is the maximal time interval in which the solution exists. In order to complete the proof of our lemma, we have to demonstrate that, when Tmax < ∞, lim sup ‖u‖L∞ ([0,t ]×[g (t ),h(t )]) = ∞. t →Tmax
Next, we take advantage of the contradiction argument. Assume that Tmax < ∞ and that lim sup ‖u‖L∞ ([0,t ]×[g (t ),h(t )]) < ∞; t →Tmax
then there exist M1 , M2 > 0 such that Tmax ≤ M1 < ∞ and ‖u‖L∞ [g (t ),h(t )] ≤ M2 < ∞ for all t ∈ [0, Tmax ). In what follows, we first prove that h′ (t ) and g ′ (t ) are uniformly bounded in (0, Tmax ), i.e., h′ (t ) ≤ M3 and −g ′ (t ) ≤ M4 for all t ∈ (0, Tmax ), with some M3 , M4 independent of Tmax . To obtain h′ (t ) ≤ M3 , we define
Ω = ΩM := {(t , x) : 0 < t < Tmax , h(t ) − M −1 < x < h(t )} and construct an auxiliary function
w(t , x) := M2 [2M (h(t ) − x) − M 2 (h(t ) − x)2 ]. We will choose M so that w(t , x) ≥ u(t , x) holds over Ω . Direct computations yield that, for (t , x) ∈ Ω ,
wt = 2M2 Mh′ (t )(1 − M (h(t ) − x)) ≥ 0, p −wxx = 2M2 M 2 , up (t , 0) ≤ M2 . It follows that
wt − dwxx ≥ 2dM2 M 2 ≥ up (t , 0) in Ω if M 2 ≥
P −1
M2
2d
. On the other hand,
w(t , h(t ) − M −1 ) = M2 ≥ u(t , h(t ) − M −1 ),
w(t , h(t )) = 0 = u(t , h(t )).
Thus, if we can choose M such that u0 (x) ≤ w(0, x) for x ∈ [h0 − M −1 , h0 ], then we can apply the maximum principle to w − u over Ω to deduce that u(t , x) ≤ w(t , x) for (t , x) ∈ Ω . It would then follow that ux (t , h(t )) ≥ wx (t , h(t )) = −2MM2 ,
h′ (t ) = −µux (t , h(t )) ≤ M3 := 2MM2 µ.
Hence, now, we only have to find some M independent of Tmax such that u0 (x) ≤ w(0, x) for x ∈ [h0 − M −1 , h0 ]. We calculate
wx (0, x) = −2M2 M [1 − M (h0 − x)] ≤ −M2 M for x ∈ [h0 − (2M )−1 , h0 ]. 4‖u0 ‖ 1 C ([0,h0 ]) (M2 )P −1 , we will have wx (0, x) ≤ u′0 (x) for x ∈ [h0 − (2M )−1 , h0 ]. Therefore, upon choosing M := max , 2d 3M2 Since w(0, h0 ) = u0 (h0 ) = 0, the above inequality implies that w(0, x) ≥ u0 (x) for x ∈ [h0 − (2M )−1 , h0 ]. Moreover, for x ∈ [h0 − M −1 , h0 − (2M )−1 ], we have
w(0, x) ≥
3 4
M2 ,
u0 (x) ≤ ‖u0 ‖C 1 ([0,h0 ]) M −1 ≤
3 4
M2 .
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Therefore, u0 (x) ≤ w(0, x) for x ∈ [h0 −M −1 , h0 ], which tells us that h′ (t ) ≤ M3 in [0, Tmax ), with M3 = 2MM2 µ independent of Tmax . Similarly, we can obtain that −g ′ (t ) ≤ M4 in [0, Tmax ) and that M4 is also independent of Tmax . Since h′ (t ) and g ′ (t ) are uniformly bounded in [0, Tmax ), we can now find L∗ > 0 depending on M1 , M2 , M3 and M4 such that ‖u(t , x)‖C 1+α [g (t ),h(t )] ≤ L∗ for all t ∈ [0, Tmax ). Using Theorem 2.1 again, we can conclude that there exists a τ > 0 depending only on M3 , M4 and L∗ such that the solution of problem (1.1) with the initial time Tmax − τ2 can be extended uniquely to the time Tmax − τ2 + τ , which is a contradiction to the hypothesis. Thus the proof is ended. In what follows, we will give the blowup result of the solution of (1.1). Theorem 3.1. Let u(t , x) be a solution of problem (1.1). Then the solution u(t , x) blows up at a finite time provided that u0 (x) is sufficiently large in a neighborhood of 0. Proof. Here, we verify the above result mainly by the maximum principle and the comparison principle. Consider the following auxiliary problem:
vt − dvxx = v p (t , 0), −h0 < x < h0 , 0 < t < T , v(t , h0 ) = v(t , −h0 ) = 0, 0 < t < T , v(0, x) = u0 (x), 0 < t < T .
(3.2)
We assert that u(t , x) ⩾ v(t , x) on [−h0 , h0 ] × (0, T ] by using the comparison principle. On the other hand, according to Theorem 2.3 in [3], we know that under the assumption that u0 (x) is sufficiently large in a neighborhood of 0, v(t , x) will cease to exist at a finite time. Therefore, the blowup result also holds for u(t , x) as u(t , x) is an upper solution, which completes the proof. Although the above theorem provides a condition to ensure the finite-time blowup, the condition for the initial function u0 (x) is very rigid. Next we attempt to find some other conditions which are more specific about the size of u0 (x). Indeed, let w(x) be the solution of the problem
−dwxx = 1, −h0 < x < h0 , w(h0 ) = w(−h0 ) = 0.
(3.3) p
Then the nonnegative steady-state solutions of (3.2) can be written as v(x) = αw(x), where α = w(0) 1−p . Theorem 3.2. If u0 (x) > (
h20
/2d)
p 1−p
−1 2
h20
2d
2d
x +
, then the solution of (1.1) blows up in finite time.
1 2 Proof. Direct calculations yield that w(x) = − x + 2d Corollary 2.4 in [3].
h20 2d
p
and α = (h20 /2d) 1−p . Hence, the desired result follows from
The next blowup result will be used to derive the global slow solution. Theorem 3.3. Let ϕ(x) be the first eigenfunction of the eigenvalue problem
−ϕxx = λ1 ϕ, −h0 < x < h0 , (3.4) ϕ(h0 ) = ϕ(−h0 ) = 0, h with ϕ(x) > 0 in [−h0 , h0 ] and −0h ϕ(x)dx = 1. Then the solution of problem (1.1) with the initial function u0 (x) in the form 0 −1 1 h0 p−1 2 of M ϕ(x) ceases to exist in finite time provided that M > λ1 . −h0 ϕ (x)dx
Proof. Consider the following auxiliary problem:
vt − dvxx = f (v(t , 0)) := v p (t , 0), −h0 < x < h0 , 0 < t < T , v(t , h0 ) = v(t , −h0 ) = 0, 0 < t < T , v(0, x) = M ϕ(x), 0 < t < T .
(3.5)
It follows that u(t , x) ⩾ v(t , x) on [−h0 , h0 ] × (0, T ] from the comparison principle. Next, we prove that v(t , x) blows up at a finite time. Multiplying (3.5) by ϕ and then integrating over [−h0 , h0 ], we obtain F ′ (t ) + λ1 F (t ) =
∫
h0
−h0
f (v(t , 0))ϕ(x)dx,
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where F (t ) =
h0
∫
v(t , x)ϕ(x)dx.
−h0
Since f is convex and ϕ is positive within (−h0 , h0 ), we have
∫
h0
f (v)ϕ(x)dx ≥ f (F (t )).
−h0
At the same time, due to the concavity of ϕ(x) over [−h0 , h0 ], it is not difficult to deduce that v(t , 0) ≥ v(t , x) on [0, T ] × [−h0 , h0 ]. Hence, we have ∫ h0 f (v(t , x))ϕ(x)dx ≥ f (F (t )). F ′ (t ) + λ1 F (t ) ≥ −h0 1 p−1
h
As −0h u0 (x)ϕ(x)dx > λ1 0 is complete.
, we have f (F (0)) ≥ λ1 F (0), which implies that v(t , x) will blow up in a finite time. The proof
Remark 3.1. From Theorem 3.3, we know that, if u0 (x) is in the form of M ϕ(x), then the solution will grow to infinity provided that M is sufficiently large. 4. The global fast solution and the slow solution In this section, we mainly investigate the long-time behavior of the global solutions of (1.1). Here, we first give the existence of a fast solution by the following theorem. Theorem 4.1 (Fast Solution). Let u be a solution of problem (1.1). If u0 is small in the following sense:
‖ u0 ‖ ∞ ⩽
1 2
1
min [d/(16h20 )] p−1 , 1/(8µ) ,
then Tmax = ∞. Moreover, h∞ < ∞, g∞ > −∞, and there exist real numbers C , β > 0 depending on u0 such that
‖u(t )‖∞ ⩽ C e−β t ,
t ⩾ 0.
(4.1)
Proof. It suffices to the construct a suitable global supersolution. Motivated by [19], we define
ϑ(t ) = 2h0 (2 − e−γ t ),
λ(t ) = −ϑ(t ),
t ⩾ 0,
V (y) = 1 − y2 ,
−1 ⩽ y ⩽ 1,
and
v(x, t ) = ε e−β t V (x/ϑ),
λ(t ) ⩽ x ⩽ ϑ(t ), t ⩾ 0,
where γ , β and ε > 0 are to be determined later. Direct computation yields
vt − dvxx − v p (t , 0) = ε e−β t [−β V − xϑ ′ ϑ −2 V ′ − dϑ −2 V ′′ − ε p−1 e−β(p−1)t ] [ ] d p−1 ⩾ ε e−β t −β + − ε , 2 8h0
for all t > 0 and λ(t ) < x < ϑ(t ). On the other hand, we can easily get ϑ ′ (t ) = 2γ h0 e−γ t > 0 and −vx (t , ϑ(t )) = 2ε e−β t /ϑ(t ). Setting γ = β =
d 2
,
16h0 1 2 p−1 and ε ⩽ ε0 = min [d/(16h0 )] , 1/(8µ) , it follows that v − dvxx − v p (t , 0) ⩾ 0, λ(t ) < x < ϑ(t ), t > 0, t′ ϑ (t ) > −µvx (t , ϑ(t )), λ′ (t ) < −µvx (t , λ(t )), t > 0, v(t , ϑ(t )) = v(t , λ(t )) = 0, t > 0, ϑ(0) = 2h0 > h0 , λ(0) = −2h0 < −h0 . 1 Assuming that ‖u0 ‖∞ ⩽ 12 min [d/(16h20 )] p−1 , 1/(8µ) and choosing ε = 2‖u0 ‖∞ , we also get u0 (x) < v(x, 0) for −h0 ⩽ x ⩽ h0 . By applying the comparison principle, one can see that h(t ) < ϑ(t ), g (t ) > λ(t ) and that u(x, t ) < v(x, t ) for g (t ) ⩽ x ⩽ h(t ), as long as u exists. In particular, it follows from the continuation property (3.1) that u exists globally.
The proof is complete.
P. Zhou et al. / Nonlinear Analysis 74 (2011) 2523–2533
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Remark 4.1. The above result indicates that the free boundaries both converge to finite limits as t tends to infinity and that the solution u(t , x) decays uniformly to 0 at an exponential rate. The solution is therefore called a fast solution compared with the case in Theorem 4.2, where the double free boundary fronts will both grow up to infinity. Before giving the existence of a global slow solution, we need the following lemma, which provides a priori estimate for the global solution. Lemma 4.1. Let u be a solution of problem (1.1) with u0 (x) in the form of δϕ(x). If Tmax = ∞, then there exists a constant C = C (‖u0 ‖C 1+α , h0 , 1/h0 ), such that sup ‖u(t , x)‖L∞ (g (t ),h(t )) ⩽ C , t ⩾0
where C remains bounded for ‖u0 ‖C 1+α , h0 , and 1/h0 bounded. Proof. First, from the local theory for problem (1.1), for each M > 1, there exists σ > 0 such that, if ‖u0 ‖C 1+α < M and 1/M < h0 < M, then ‖u(t , x)‖L∞ < 2M on [0, σ ]. In what follows, we use the contradiction argument. If the result is false, then there exists an M > 0 and a sequence of global solutions (un , gn , hn ) of (1.1), such that 1/M < gn (0) < M ,
‖un (0, x)‖C 1+α [−h0 ,h0 ] < M
and sup ‖un (t , x)‖L∞ (gn (t ),hn (t )) → ∞ as n → ∞. t ⩾0
For all large n, there exist tn ⩾ σ and xn ∈ (gn (t ), hn (t )) such that 3 4
sup ‖un (t , x)‖L∞ (gn (t ),hn (t )) = un (tn , xn ). t ⩾0
We claim that xn = 0. In fact, we set v(t , x) = ux (t , x). Since now u0 = λϕ is symmetric and concave on [−h0 , h0 ], we have h(t ) = −g (t ), and, for any t ∈ [0, ∞), the corresponding solution is also symmetric over (g (t ), h(t )). Hence, we can deduce that v(t , 0) = ux (t , 0) = 0. Then v(t , x) satisfies
v − dvxx = 0, t > 0, 0 < x < h(t ), t v(t , h(t )) = ux (t , h(t )) ≤ 0, t > 0, v(t , 0) = ux′ (t , 0) = 0, t > 0, v(0, x) = u0 (x) ≤ 0, 0 ≤ x ≤ h0 .
(4.2)
By the maximum principle, we can deduce that v(t , x) ≤ 0 for x ∈ (0, h(t )). Similarly, we can obtain that v(t , x) ≥ 0 for x ∈ (−g (t ), 0). Therefore, u(t , x) ≤ u(t , 0) over [0, +∞) × [g (t ), h(t )]. Thus, we have xn = 0, which implies that sup ‖un (t , x)‖L∞ (gn (t ),hn (t )) = un (tn , 0) , ϱn . t ⩾0
−(p−1)/2
We define λn = ϱn ; then it is evident to see that λn → 0 as n → ∞. We extend un (t , ·) by 0 on (hn (t ), ∞) and (−∞, gn (t )) and define the rescaled function
vn (τ , y) = λ2n un (tn + λ2n τ , λn y)
(4.3)
2 for (τ , y) ∈ Dn = {(τ , y) : −λ− n tn ⩽ τ ⩽ 0 and − ∞ < y < ∞}. Also, we denote 1 2 y 1 = λ− n g (tn + λn τ ),
1 2 y2 (τ ) = λ− n h(tn + λn τ )
and 2 Dn = {(τ , y) : −λ− n tn ⩽ τ ⩽ 0 and y1 (τ ) ⩽ y < y2 (τ )},
which corresponds to the domain {g (t ) < x < h(t )}. The function vn satisfies vn (0, 0) = 1, 0 ⩽ vn ⩽
∂τ vn − d∂y2 vn = vnp (τ , 0),
4 3
and
(y, τ ) ∈ Dn .
(4.4) p Lloc
Similarly to Lemma 2.1 in [1], there exists a subsequence {vnk } of {vn } such that {vnk } converges in ((−∞, 0] × (−∞, +∞)) to a function w(τ , y) ∈ Lploc ((−∞, 0] × (−∞, +∞)), and {vnk (0, y)} converges in Cloc (−∞, +∞) to a function z (y) ∈ C (−∞, +∞), which satisfies z (0) = 1. Moreover, similarly to Lemmas 2.2 and 2.3 in [1], wt = 0 in D′ ((−∞, 0) × (−∞, +∞)), and there is a function w(y) ⩾ 0 which is bounded, continuous on (−∞, ∞), and satisfies that −wyy = wp (0); hence w is concave. Therefore, w ≡ 0, which leads to a contradiction to the fact that w(0) = z (0) = 1. This completes the proof of Lemma 4.1.
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The above result indicates that the global solution is uniformly bounded while the following lemma will point out that the global solution decays uniformly to 0. Lemma 4.2. Under the same condition as that in Lemma 4.1, the solution satisfies lim ‖u(t , x)‖L∞ (g (t ),h(t )) = 0.
t →+∞
Proof. Assume that k := lim supt →+∞ ‖u(t , x)‖L∞ (g (t ),h(t )) > 0 by contradiction. It follows from Lemma 4.1 that k < +∞. Let t0 > 0 be such that sup ‖u(t , x)‖L∞ (g (t ),h(t )) ⩽
[t0 ,+∞)
3 2
k.
Then there exists a sequence tn → +∞ such that ‖u(tn , x)‖L∞ (g (tn ),h(tn )) ⩾
3 k. 4
As in Lemma 4.1, we can obtain
‖u(tn , x)‖L∞ (g (tn ),h(tn )) = u(tn , 0) and then define λn = u−(p−1)/2 (tn , xn ). We extend u(t , ·) by 0 on (−∞, g (t )) and (h(t ), ∞) and define the rescaled function 2
vn (τ , y) = λnp−1 u(tn + λ2n τ , λn y)
(4.5)
2 for (τ , y) ∈ Dn = {(τ , y) : λ− n (t0 − tn ) ⩽ τ ⩽ 0 and − ∞ < y < ∞}. Also, we denote 2 1 y1 (τ ) = λ− n g (tn + λn ),
1 2 y2 (τ ) = λ− n h(tn + λn τ )
and 2 Dn = {(τ , y) : λ− n (t0 − tn ) ⩽ τ ⩽ 0 and y1 ⩽ y < y2 (τ )}.
Therefore, the function vn satisfies vn (0, 0) = 1, 0 ⩽ vn ⩽ 2 ∈ Dn , and
∂τ vn − d∂y2 vn = vnp (τ , 0),
(τ , y) ∈ Dn .
(4.6)
Similarly to Lemmas 2.1–2.3 in [1], we have obtained a function w(y) ⩾ 0, bounded and continuous on (−∞, ∞), which satisfies that −wyy = w p (0). Therefore, w ≡ 0, which contradicts the fact that w(0) = 1 since vn (0, 0) = 1. The proof is ended. Theorem 4.2 (Slow Solution). Let ϕ(x) be the first eigenfunction of the eigenvalue problem (3.4). Then there exists λ > 0 such that the solution of (1.1) with initial data u0 = λϕ is a global slow solution, which satisfies that h∞ = −g∞ = ∞. Proof. In order to emphasize the dependence of u on the initial data when necessary, we denote the solution of (1.1) by u(u0 ; ·). So do g (t ), h(t ), g∞ , h∞ and the maximal existence time T . We first claim that when u0 = λϕ we have h(t ) = −g (t ), since the initial function is symmetric on [−h0 , h0 ], which implies that h(t ) and g (t ) are both finite or infinite at the same time. Motivated by [1], we define
Σ = {λ > 0 : T (λϕ) = ∞ and h∞ (λϕ) < ∞}. According to Theorem 4.1, we know that λ ∈ Σ if λ is small, so Σ is not empty. In contrast, when λ is sufficiently large, it follows from Theorem 3.3 that the corresponding solution will blow up, i.e., T (λϕ) < ∞; hence Σ is bounded. Let
λ∗ = sup Σ ∈ (0, ∞),
v = u(λ∗ ϕ; ·),
σ = h(λ∗ ϕ; ·) and τ = T (λ∗ ϕ).
First, we claim that τ = ∞. In fact, by continuous dependence (see [20,21]), for any fixed t ∈ [0, τ ), u(λϕ; t , x) converges to v(t , x) in L∞ (−∞, ∞) and h(λϕ; t ) → σ (t ) as λ ↑ λ∗ . Here we extend u(t , x) by 0 for x ∈ (−∞, g (t )) (h(t ), ∞). It follows from Lemma 4.1 that ‖v(t )‖∞ ≤ C for all t ∈ [0, τ ), because T (λϕ) = ∞ for all λ ∈ (0, λ∗ ). Thus we have τ = ∞, since nonglobal solutions should satisfy lim supt →T ‖u(t , x)‖L∞ (g (t ),h(t )) = ∞. Next we claim that σ∞ = ∞. In what follows, we use a contradiction argument. Assume thatσ∞ < ∞. Since ‖v(t )‖∞
→ 0 as t → ∞ by Lemma 4.2, we can choose t0 sufficiently large such that ‖v(t0 )∞ ‖ <
1 2
1
min [d/(16h20 )] p−1 , 1/(8µ) .
By continuous dependence, we can deduce that
‖u(λφ; t0 , x)‖L∞ (g (t0 ),h(t0 )) ⩽
1 2
1
min [d/(16h20 )] p−1 , 1/(8µ)
for λ > λ∗ sufficiently close to λ∗ . But this implies that h∞ (λφ) < ∞ by Theorem 4.1, which is a contradiction to the definition of λ∗ . The proof is complete.
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References [1] M. Fila, P. Souplet, Existence of global solutions with slow decay and unbounded free boundary for a superlinear Stefan problem, Interfaces Free Bound. 3 (2001) 337–344. [2] H. Ghidouche, P. Souplet, D. Tarzia, Decay of global solutions, stability and blow-up for a reaction–diffusion problem with free boundary, Proc. Amer. Math. Soc. 129 (2001) 781–792. [3] J.M. Chadam, A. Peirce, H.M. Yin, The blowup property of solutions to some diffusion equations with localized nonlinear reactions, J. Math. Anal. Appl. 169 (1992) 313–328. [4] K. Bimpong-Bota, P. Ortoleva, J. Ross, Far-from-equilibrium phenomena at local sites of reaction, J. Chem. Phys. 60 (1974) 3124–3133. [5] P. Ortoleva, J. Ross, Local structures in chemical reactions with heterogeneous catalysis, J. Chem. Phys. 56 (1972) 4397. [6] J. Crank, Free and Moving Boundary Problem, Clarendon Press, Oxford, 1984. [7] J. Goodman, D.N. Ostrov, On the early exercise boundary of the American put option, SIAM J. Appl. Math. 62 (2002) 1823–1835. [8] L.S. Jiang, B.J. Bian, F.H. Yi, A parabolic variational inequality arising from the valuation of fixed rate mortgages, European J. Appl. Math. 16 (2005) 361–383. [9] X.F. Chen, A. Friedman, A free boundary problem arising in a model of wound healing, SIAM J. Math. Anal. 32 (2000) 778–800. [10] V. Guyonne, L. Lorenzi, Instability in a flame ball problem, Discrete Contin. Dyn. Syst. Ser. B 7 (2007) 315–350. [11] A. Friedman, Free boundary problems associated with multiscale tumor models, Math. Model. Nat. Phenom. 4 (2009) 134–155. [12] D. Hilhorst, M. Mimura, R. Schatzle, Vanishing latent heat limit in a Stefan-like problem arising in biology, Nonlinear Anal. RWA 4 (2003) 261–285. [13] K.I. Kim, Z.G. Lin, A free boundary problem for a parabolic system describing an ecological model, Nonlinear Anal. RWA 10 (2009) 428–436. [14] K.I. Kim, Z.G. Lin, Z. Ling, Global existence and blowup of solutions to a free boundary problem for mutualistic model, Sci. China Math. 53 (2010) 2085–2095. [15] Z.G. Lin, A free boundary problem for a predator–prey model, Nonlinearity 20 (2007) 1883–1892. [16] M. Mimura, Y. Yamada, S. Yotsutani, Free boundary problems for some reaction–diffusion equations, Hiroshima Math. J. 17 (1987) 241–280. [17] O.A. Ladyzenskaja, V.A. Solonnikov, N.N. Ural’ceva, Linear and Quasilinear Equations of Parabolic Type, Amer. Math. Soc., Providence, RI, 1968. [18] Y.H. Du, Z.G. Lin, Spreading-vanishing dichotomy in the diffusive logistic model with a free boundary, SIAM J. Math. Anal. 42 (2010) 377–405. [19] R. Ricci, D.A. Tarzia, Asymptotic behavior of the solutions of the dead-core problem, Nonlinear Anal. 13 (1989) 405–411. [20] A. Fasano, M. Primicerio, Free boundary problems for nonlinear parabolic equations with nonlinear free boundary conditions, J. Math. Anal. Appl. 72 (1979) 247–273. [21] P. Souplet, Stability and continuous dependence of solutions to one-phase Stefan problems for semilinear parabolic equations, Port. Math. 59 (2002) 315–323.