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A remark on the existence and uniqueness of solutions of a semilinear parabolic equation
Nonlinear Analysis 50 (2002) 425 – 432
Short communication
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A remark on the existence and uniqueness of solutions of a semilinear parabolic equation Tor A. Kwembe∗ Department of Mathematics and Computer Science, Chicago State University, 9501 South King Drive, Chicago, IL 60628, USA Received 15 March 2000; accepted 18 October 2000
Keywords: Convergence; Embedding; Maximal; Operator; Pointwise; Potential; Regularity; Semilinear; Weak
1. Introduction In this paper, we consider an initial value semilinear parabolic equation: 9u(x; t) − 5u(x; t) = u+1 (x; t); (x; t) ∈ Rn × (0; ∞); ¿ 0; 9t u(x; 0) = f(x); x ∈ Rn :
(1) (2)
This problem has been studied extensively in the past in [4 –8,11,14,16,17], for various restrictions on . In [4,5], Fujita who >rst studied the problem, proved that if n=2 ¡ , then for any smooth initial data f ¿ 0, f not identically equal to zero, the solution u must blow up in a >nite time regardless of how small f is. He also proved the existence of global solutions for the case n=2 ¿ and a small initial data. Kobayashi et al. [8] and Weissler [14] independently proved that for n=2 = , the solution blows up. Using the decay rates of the L∞ norm, Klainerman [7] proved the global existence of smooth solutions for n=2 ¿ 1=(1 + 1=). By the same method, Zheng and Shen [17] and Ponce [11] independently obtained solutions on the relations between and n. They proved that for f ∈ H k ∩ W k; 1 suDciently small such that k ¿ n + 4 and n=2 ¿ , problem (1) – (2) admits a unique global small solution. In both papers, the method of decay rates of the L2 and L∞ norms of the solution were used. Later on, Zheng [16] ∗ Fax: +1-773-995-3767. E-mail address: [email protected] (T.A. Kwembe).
c 2002 Elsevier Science Ltd. All rights reserved. 0362-546X/02/$ - see front matter PII: S 0 3 6 2 - 5 4 6 X ( 0 1 ) 0 0 7 7 1 - 4
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used the method of global iteration combined with that of decay rates to re>ne the results of their earlier work. In this paper, though, I have employed the application of potential operators theory to study the weak solutions of problem (1) – (2) when the initial data f is in some Lp space. On using the techniques employed in [1–3,9,10], I have established the existence and uniqueness of maximal solutions when n=2 ¿ ( + 1)=; n ¿ 2 and the initial data is very small in the norm of de>nition. The paper also considers regularity results of maximal solutions and pointwise a.e. convergence of solutions in some Lp spaces, where p = n=2. Throughout the paper, x = (x1 ; : : : ; x n ) will denote a point in Rn : |x| = (x12 + · · · + 2 1=2 x n) will take the usual meaning of the distance of the point x from the origin. Lp (Rn ); 1 6 p ¡ ∞, will denote the space of Lebesgue measurable functions over Rn called the Lp space, and for all f ∈ Lp (Rn ), ||f||p will denote the Lp norm of the function f de>ned on Rn . We also de>ne u∗ (x) = supt¿0 |u(x; t)| to be the maximal function of u(x; t) over t ¿ 0 de>ned on Rn . In this case ||u∗ ||p = [ Rn (supt¿0 |u(x; t)|)p d x]1=p will denote the Lp norm of the function u∗ (x) on Rn . In order to establish the pointwise convergence result, we will consider the solution of the problem in the in>nite cylinder ST = Rn × [0; T ) for the initial data f ∈ Lp (Rn ); p =n=2, n=2 ¿ ( + 1)=. Under this consideration, we introduce the norm ||u∗ ||p (T ) = [ Rn (sup0¡t¡T |u(x; t)|)p d x]1=p ; p = n=2, n=2 ¿ (+1)=, where we now de>ne u∗ (x) = sup0¡t¡T |u(x; t)|. These norms have been introduced in [1,2,9]. If we let W (x; t) be the fundamental solution of the diKusion equation ut = 5u, then it is well known that a formal integral solution of (1) – (2) is given by t W (x − y; t − )u+1 (y; ) dy d + W (x − y; t)f(y) dy: (3) u(x; t) = 0
Rn
Rn
If we further let be the perturbed nonlinear operator de>ned by t W (x − y; t − )u+1 (y; ) dy d + W (x − y; t)f(y) dy: (4) (u)(x; t) = 0
Rn
Rn
Then by using the method of successive approximations we showed that the iterations uk+1 = (uk ) converges in the norm of de>nition to the solution of (3) and that the mapping is a contraction mapping a ball of radius say s0 into itself. Throughout this paper, we consider a solution u of (3) for all t ¿ 0 to be a weak solution of (1) – (2) whenever the integrals involved exist in the Lebesgue sense for all values of t ¿ 0 over Rn . The instrument for obtaining results in this paper is the Sobolev embedding Theorem 1 whose proof can be found in [15]. Theorem 1. Let 0 ¡ ¡ n; 1 6 p ¡ q ¡ ∞; 1=q = 1=p − =n. (a) If f ∈ Lp (Rn ); then (I f)(x) = [()]−1 Rn |x − y|−n f(y) dy; () =
[ 3=2 2 (=2)] (n=2 − =2)
converges absolutely for almost every x. (b) If; in addition; 1 ¡ p; then ||(I f)||q 6 AP; q ||f||p .
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In what follows, we shall consider the following standard Solonnikov estimates of the fundamental solution W which are given in [2,3,9,10,12,13]. |W (x; t; y; )| 6 c(|x − y| + |t − |1=2 )−n ;
(5)
where c ¿ 0 is a constant and n is the space dimension. The estimate for the spatial derivatives of W is given by |Dx" W (x; t; y; )| 6 C(|x − y| + |t − |1=2 )−n−|"| ;
(6)
where the derivatives exist in the distribution sense, and " = ("1 ; : : : ; "n ), |"| = "1 + · · · + "n , 9 9"1 9"2 9"n 9 9 " ; Dx = : ··· : · · · "n : Dx = 9x1 9x2 9x n 9x1"1 9x2"2 9x n We will also consider the Hardy–Littlewood Maximal function f∗ of a function f ∗ de>ned as f (x) = sup1=|Q| Q |f(y)| dy, where the supremum is taken over all cubes Q with center at x and edges parallel to the coordinate axes. Then it is proven in [15] that Theorem 2. Let 1 ¡ p 6 ∞ and f ∈ Lp (Rn ). Then f∗ ∈ Lp (Rn ) and ||f∗ ||p 6 C||f||p where C depends only on n and p. We shall refer to Theorem 2 as the maximal Theorem. We shall also consider the following known properties of W . (i) Rn W (x; t) d x = 1 (ii) supt¿0 Rn W (x − y; t)f(y) dy 6 cf∗ (x), where c is independent of f |"| t f(x−y) dy ∗ (iii) supt¿0 Rn |y| n+|"| +t n+|"| 6 Cf (x). (iv) W ∈ C0∞ . From property (iii), we have the well known result whose proof can be found in [15]. Theorem 3. Let H|"| (f)(x) = p
Rn
t |"| f(y) dy : |x − y|n+|"| + t n+|"|
n
If f ∈ L (R ); 1 6 p ¡ ∞; and |"| ¿ 0; then H|"| (f) ∈ Lp (Rn ) and ||H|"| (f)||p 6 C||f||p ; where C is a constant independent of f. 2. Estimates for the integral equation and xed-point properties Lemma 4. Let u(x; t) be a weak solution of problem (1) – (2) satisfying the integral equation (3). Then; the nonlinear transformation in (4) maps Ln=2 (Rn∗ ) to Ln=2 (Rn∗ ) and ||(u∗ )||n=2 6 C1 ||u∗ ||+1 n=2 + C2 ||f||n=2 ; n=2 ¿ + 1=; n ¿ 2; ¿ 0; where by n=2 n n=2 L (R∗ ) we mean the L norm of the maximal function u∗ (x) over Rn and the constants C1 ; C2 depends only on and n.
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t Proof. From (4) we have (u)(x; t) = 0 Rn W (x−y; t−)u+1 (y; ) dy d+ Rn W (x − y; t)f(y) dy. Then on utilizing estimate (5) we have t |u(y; )|+1 dy d |(u)(x; t)| 6 c + W (x − y; t)f(y) dy : 1=2 n Rn (|x − y| + |t − | ) Rn 0 On taking the supremum over t ¿ 0 of both sides of the inequality we have W (x − y; t)f(y) dy ; (u∗ )(x) 6 c C0 [u∗ (y)]+1 dy=|x − y|n−2 + sup Rn
t¿0
Rn
∞ √ where C0 is such that 0 ds=(1 + s)n 6 C0 . Then for p ¿ 1, we take the Lp norm of both sides and we use the Minkowski’s inequality on the right-hand side. For the ensuing >rst term on the right-hand side, we apply the Sobolev embedding Theorem 1. That is for p ¿ 1, we can >nd a q ¿ + 1 such that if, for 1=p = ( + 1)=q − 2=n; n ¿ 2 and p ¿ q= + 1 ¿ 1, [u∗ (x)]+1 ∈ Lq=+1 (Rn ); then [u∗ (y)]+1 dy ∗ +1 ∗ +1 n |x − y|n−2 6 C(q)||[u ] ||q=+1 = C(q)||u ||q : R P For the ensuing last term of the right-hand side of the inequality, property (ii) combined with Theorem 2 gives ||supt¿0 | Rn W (x − y; t)f(y) dy| ||p 6 C(p)||f||p . A combina+ C(P)||f||P . If we now require that tion of these gives ||(u∗ )||p 6 cC0 C(q)||u∗ ||+1 q p = q ¿ +1 in the equation 1=p = (+1)=q−(2=n), then we see that p = n=2 ¿ +1. From which we see that n=2 ¿ ( + 1)=. This concludes the proof of the lemma. Lemma 5. The nonlinear operator (u)(x; t) maps the ball {||u∗ ||n=2 6 s0 } into itself; if s0 is the smallest positive root of the equation s = C1 s+1 + C2 ||f||n=2 provided that C1 ; C2 and ||f||n=2 are chosen such that the graph of the line y = s remains underneath the graph of y = C1 s+1 + C2 ||f||n=2 ; and C2 ||f||n=2 6 s0 in 0 6 s 6 s0 . If C1 ( + 1)s1 ¡ 1; where 0 ¡ s1 ¡ s0 ; then is a contraction mapping in the ball of radius s0 . Proof. That maps the ball ||u∗ ||n=2 6 s0 into itself follows directly from Lemma 4. From Eq. (3) we have that u(x; t) = (u)(x; t). Therefore, u∗ (x) 6 (u∗ )(x). Hence for n=2 ¿ ( + 1)=; ¿ 0; we have by Lemma 4 that ||u∗ ||n=2 6 C1 ||u∗ ||+1 n=2 + C2 ||f||n=2 ; where Ci ; i = 1; 2 are constants depending only on and n. Let y = C1 s+1 +C2 ||f||n=2 . Then consider the function g(s) = C1 s+1 − s + C2 ||f||n=2 . By the Newton’s method of successive approximations, we can generate a succession of points s1 ; s2 ; : : : ; sk ; : : : 1 given by sk+1 = sk − g(sk )=g (sk ), k = 0; 1; 2; : : : ; sk = [C1 (+1)] 1= such that the sequence {sk } converges to a limit s0 ¿ 0, such that g(s0 ) = 0. Hence the function y intersect the line y = s at the point (s0 ; y(s0 )). Furthermore for C2 ||f||n=2 1, and small values of s, y ¡ s. Therefore, we can >nd a point say s1 with s0 ¡s1 such that a sequence of Newton’s approximation of g in the neighborhood of s1 converges to s1 and g(s1 ) = 0. Since y ¿ 0 in (s0 ; s1 ); y is convex in (s0 ; s1 ); and so in (0; s0 ); s 6 y and C2 ||f||n=2 6 y(s0 ). Hence, the line y = s remains underneath the graph of y = C1 s+1 +
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C2 ||f||n=2 and the graph of y = C1 s+1 + C2 ||f||n=2 remains underneath the line segment joining the points (0; C2 ||f||n=2 ) and (s0 ; s0 ). Thus y is convex in (0; s0 ) with respect to this line segment and hence we can >nd two points a, b in (0; s0 ) such that |y(b) − y(a)| 6 y ())|b − a| for some ) in (0; s0 ). Since g is a decreasing function on (0; s0 ), we have that g ¡ 0 for all s ∈ (0; s0 ). Thus, C1 ( + 1)s − 1 ¡ 0; s ∈ (0; s0 ). Thus, C1 ( + 1)s ¡ 1. Hence, the lemma follows if we replace s with ||u∗ ||n=2 and y with .
3. Main results We now state the main theorems of the paper. Theorem 6. Let u(x; t) be a weak solution of problem (1) – (2) satisfying the integral equation (3). Let f ∈ Ln=2 (Rn ); n=2 ¿ ( + 1)=; ¿ 0; and n ¿ 2 such that for a small *0 ¿ 0; ||f||n=2 ¡ *0 ; then; (i) problem (1) – (2) has at most one weak solution; and (ii) for all u(x; t) satisfying the integral equation (3); ||u∗ ||n=2 is 6nite. Proof. The proof follows from Lemmas 4 and 5. Theorem 7. Let u(x; t) be a weak solution of problem (1) – (2). If f ∈ Ln=2 (Rn ); n=2 ¿ ( + 1)=; ¿ 0; and n ¿ 2; then; (i) ||u∗ ||n=2 is 6nite and (ii) the function u(x; t) converges a.e. to f(x) in Ln=2 (Rn ) as t goes to zero. We will present the proof of Theorem 7 along the lines of the pointwise convergence argument in [1]. But, before we establish the proof of Theorem 7, we shall consider the following lemma: t Lemma 8. Let V (x; t) = 0 Rn W (x − y; t − )u+1 (y; ) dy d. If u ∈ Ln=2 (Rn × R+ ); n=2 ¿ ( + 1)=, ¿ 0. Then ||V ∗ ||n=2 (T ) 6 C||u∗ ||+1 n=2 (T ). Proof. On utilizing estimate (5) on V , we have t |u(y; )|+1 dy d |V (x; t)| 6 : 1=2 n 0 Rn (|x − y| + |t − | ) On taking the supremum over 0 ¡ t ¡ T on both sides we have sup0¡t¡T |V (x; t)| 6 T √ C0 Rn ([u∗ (y; T )]+1 dy)=(|x − y|n−2 ), where C0 is such that 0 ds=(1 + s)n 6 C0 . Then, if we assume that [u∗ (y; T )]+1 ∈ Lp=+1 (Rn ); we apply the Sobolev embedding Theorem for p ¿ + 1 such that 1=p = ( + 1)=p − 2=n; n ¿ 2, we get p ||sup0¡t¡T |V (x; t)|||p 6 C ||u∗ (T )+1 ||p=+1 = C(; n)||u∗ ||+1 n=2 (T ) +1 and the lemma follows. Furthermore, property (ii) and the maximal theorem tells us that Rn W (x−y; t)f(y) dy belongs to Ln=2 (Rn ) and ||sup0¡t¡T | Rn W (x−y; t)f(y) dy|
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T.A. Kwembe / Nonlinear Analysis 50 (2002) 425 – 432
||n=2 6 C||f||n=2 . Thus, the weak solution u of problem (1) – (2) belongs to the class of norms ||u∗ ||n=2 (T ). Proof of Theorem 7. By property (i), we have that u(x; t) − f(x) = V (x; t) + Rn W (x − y; t)[f(y) − f(x)] dy. On taking the Ln=2 (Rn ) norm of both sides, and then the limit as t approaches zero on both sides, we have the desired result after the following computation: By the continuity of f and a well known result in [15], we have that || Rn W (x − y; t)[f(y) − f(x)] dy||n=2 tends to zero with t. Also as a consequence of Lemma 8 and the fact that |V (x; t)| 6 C Rn ,(x − y; t)|x − y|−n+2 [u(y; T )]+1 dy, where t ,(y; t) = 0 |y|−2 (1 + |y|−1 s1=2 )−n ds, we see that V (x; t) tends to zero in Ln=2 norm as t tends to zero. Hence u(x; t) → f(x) a.e. in Ln=2 (Rn ) as t → 0.
4. Regularity results We shall employ the following standard result for the convolution derivative whose proof can be found in [15], to establish regularity results for the maximal solution of problem (1) – (2). Lemma 9. If 1 6 p ¡ ∞; f ∈ Lp (Rn ) and W ∈ C0∞ ; then f ∗ W ∈ C ∞ and D" (f ∗ W )(x) = (f ∗ D" W )(x). Theorem 10. Let |"| 6 M; be the order of the spatial distributional derivative of the maximal solution. Let u(x; t) be the weak solution of problem (1) – (2) given by (3). If n + |"|=2 ¿ + 1=; and f ∈ L((n+|"|))=2 (Rn ); then D" u(x; t) ∈ L((n+|"|))=2 (Rn∗ ) in the (+1) class of Lp (Rn∗ ) norm and ||D" u∗ ||(n+|"|)=2 6 C1 ||u∗ ||((n+|"|))=2 + C2 ||f||((n+|"|))=2 . t Proof. Let u(x; t) = 0 Rn W (x − y; t − )u+1 (y; ) dy d + Rn W (x − y; t)f(y) dy. t Then by Lemma 9, we have D" u(x; t) = 0 Rn (Dx" W (x − y; t − ))u+1 (y; ) dy d + Rn (Dx" W (x − y; t))f(y) dy. On using estimate (6), we have t |u(y; )|+1 dy d |f(y)| dy |D" u(x; t)| 6 C + C : 1=2 )n+|"| (|x − y| + |t − | (|x − y| + t 1=2 )n+|"| n n 0 R R Since (|x − y| + t 1=2 )n+|"| ¿ t −|"|=2 (|x − y|n+|"| + t (n+|"|)=2 ), we have that t |u(y; )|+1 dy d t |"|=2 |f(y)| dy |D" u(x; t)| 6 C + C : 1=2 n+|"| n+|"| + t (n+|"|)=2 0 Rn (|x − y| + |t − | ) Rn |x − y| Therefore, on taking the supremum of both sides over t ¿ 0, we have "
sup |D u(x; t)| 6 C0 C t¿0
Rn
[u∗ (y)]+1 dy + C sup |x − y|n+|"|−2 t¿0
Rn
t |"|=2 |f(y)| dy |x − y|n+|"| + t (n+|"|)=2
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∞ √ where C0 is such that 0 ds=(1 + s)n+|"| 6 C0 . Then we take the Lp (Rn ) norm of both sides using the Minkowski’s inequality on the right-hand side. Then we have two terms on the right-hand side to evaluate. Concerning the >rst term, if p, q are such that q ¿ + 1, and 1=p = + 1=q − 2=n + |"|; n + |"| ¿ 2, then we apply the Sobolev embedding theorem on the >rst term if [u∗ (y)]+1 ∈ Lq=+1 (Rn∗ ). We then apply property (iii) and the maximal Theorem 1 on the last term. Then, we have ||D" u∗ ||p 6 C0 CK(q=+1) ||u∗ ||+1 q +CK(p)||f||p . If we let p = q ¿ +1 in 1=p = ((+1)=q)−(2=(n+|"|)), then we have p = (n+|"|)=2 ¿ +1. From which we have that n+|"|=2 ¿ +1=. Hence, (+1) we have the required result of ||D" u∗ ||(n+|"|)=2 6 C1 ||u∗ ||(n+|"|)=2 + C2 ||f||(n+|"|)=2 ; n + |"|=2 ¿ + 1= and ¿ 0, if we take C1 = C0 CK((n + |"|)=2( + 1)), and C2 = CK((n + |"|)=2). Remark. The use of the above method for the case of classical solutions studied by some of the authors mentioned in the introduction may not hold. Especially when boundary value problems are introduced. For, if one consider the simple case (i) 9u=9t − 5u = u+1 ; (x; t) ∈ D × (0; ∞) ⊂ Rn × R+ ; (ii) u(x; 0) = f(x); x ∈ D, and (iii) 9u 9n = 0; (x; t)∈ 9D × (0; ∞), the existence of solution for all time t ¿ 0 depends on the sign of ( D f(x) d x) , ¿ 0. Since, if we let V (t) = D u(x; t) d x, and inte+1 grate (i) over D using (ii) and (iii), we have (iv) dV (t)=dt = D u (x; t) d x, and (v) V (0) = D u(x; 0) d x = D f(x) d x. On applying the Holder’s inequality of exponents ( + 1) and + 1=, we have that V (t) 6 |D| D u+1 (x; t) d x. Hence dV (t)=dt ¿ V (t)+1 =|D| , where |D| is the volume of the region D. Now, if we let y(t) be the solution of the Cauchy problem for the Ricatti equation (vi) dy(t)=dt = y(t)+1 =|D| , and (vii) y(0) = D f(x) d x ¿ 0. Then clearly we have that V (t) ¿ y(t). But, then f(x) d x D y(t) =
1= ; ¿ 0: 1 − =|D| D f(x) d x t If ( D f(x) d x) ¿ 0, then y(t) → ∞ as t → |D| =( D f(x) d x) . Hence in this case the solutions blows up at t0 = |D| =( D f(x) d x) . Hence, we can only obtain local solution on the interval [0; t0 ). Therefore, V (t) must blow up in a >nite time, and consequently so does the solution of (i) – (iii). However, if ( D f(x) d x) ¡ 0, then the solution exists for all time t ¿ 0. Thus, in this situation one has to be more careful t in establishing the aboveresults for the representative solution u(x; t) = 0 D W (x − y; t − )u+1 (y; ) dy d + D W (x − y; t)f(y) dy. An adjustment to maximal functions of u over an interval [0; t0 ) can be made if we de>ne u∗ (x) = sup06t¡t0 |u(x; t)| and then follow the treatment in establishing pointwise convergence. Then the existence and uniqueness as well as pointwise convergence results will hold.
References [1] C.P. CalderMon, Initial values of solutions of the Navier–Stokes equations, Proc. AMS 117 (3) (1993) 761–766. [2] C.P. CalderMon, T.A. Kwembe, On the classical trapping problem, Math. Biosc. 102 (2) (1990) 183–190.
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[3] E.B. Fabes, B.F. Jones, N.M. Riviere, The initial value problem for the Navier–Stokes equations with data in Lp , Arch. Rational Mech. Anal. 45 (1972) 222–240. [4] H. Fujita, On the blowing up of solutions of the Cauchy problem for ut = 5u + u+1 , J. Fac. Sci. Univ. Tokyo, Sect. 1A 13 (1966) 109–124. [5] H. Fujita, On some nonexistence and non-uniqueness theorems for nonlinear parabolic equations, Proc. Sympos. Pure Math. 18 (1970) 105–113. [6] K. Hayakawa, On nonexistence of global solutions of some semilinear parabolic diKerential equations, Proc. Japan Acad. 49 (1973) 503–505. [7] S. Klainerman, Long-time behavior of solutions to nonlinear evolution equations, Arch. Rational Mech. Anal. 78 (1982) 73–98. [8] K. Kobayashi, T. Sirao, H. Tanaka, On the growing up problem for semilinear heat equations, J. Math. Soc. Japan 29 (1977) 407–424. [9] T.A. Kwembe, The initial boundary value problems for the semilinear diKusions with data in Lp spaces, J. Appl. Anal. 72 (1–2) (1999) 205–228. [10] J.E. Lewis, The initial-boundary value problem for the Navier–Stokes equations with data in Lp , Indiana Univ. Math. J. 23 (8) (1973) 739–761. [11] G. Ponce, Global existence of solutions to a class of nonlinear evolution equations, Nonlinear Anal. 9 (1985) 71–92. [12] V.A. Solonnikov, Existence of solutions of non-stationary linearized systems of Navier–Stokes equations, Trudy Mat. Inst. Steklov. 70 (1964) 213–317 [in English: AMS Transl. Ser. II (1975) 1–116]. [13] V.A. Solonnikov, A priori estimates for second-order parabolic equations, Trudy Mat. Inst. Steklov, 70 (1964), 133–212 [in English: AMS Transl. 2 (65) (1967) 51–137]. [14] F.B. Weissler, Existence and non-existence of global solutions for a semilinear heat equation, Israel J. Math. 38 (1–2) (1981) 29–40. [15] R.L. Wheeden, A. Zygmund, Measure and Integration: An Introduction to Real Analysis. Monographs and Textbooks in Pure and Applied Mathematics, No. 43, Marcel Dekker, New York, 1977. [16] S. Zheng, Nonlinear Parabolic Equations and Hyperbolic–Parabolic Coupled Systems, Pitman Monographs and Surveys in Pure and Applied Mathematics Vol. 76, Longman–Wiley, England, New York, 1995. [17] S. Zheng, W. Shen, Global existence of initial boundary value problem for Fitzhugh–Nagumo equations, Sci. Bull. 9 (1984) 16–20.
Short communication
www.elsevier.com/locate/na
A remark on the existence and uniqueness of solutions of a semilinear parabolic equation Tor A. Kwembe∗ Department of Mathematics and Computer Science, Chicago State University, 9501 South King Drive, Chicago, IL 60628, USA Received 15 March 2000; accepted 18 October 2000
Keywords: Convergence; Embedding; Maximal; Operator; Pointwise; Potential; Regularity; Semilinear; Weak
1. Introduction In this paper, we consider an initial value semilinear parabolic equation: 9u(x; t) − 5u(x; t) = u+1 (x; t); (x; t) ∈ Rn × (0; ∞); ¿ 0; 9t u(x; 0) = f(x); x ∈ Rn :
(1) (2)
This problem has been studied extensively in the past in [4 –8,11,14,16,17], for various restrictions on . In [4,5], Fujita who >rst studied the problem, proved that if n=2 ¡ , then for any smooth initial data f ¿ 0, f not identically equal to zero, the solution u must blow up in a >nite time regardless of how small f is. He also proved the existence of global solutions for the case n=2 ¿ and a small initial data. Kobayashi et al. [8] and Weissler [14] independently proved that for n=2 = , the solution blows up. Using the decay rates of the L∞ norm, Klainerman [7] proved the global existence of smooth solutions for n=2 ¿ 1=(1 + 1=). By the same method, Zheng and Shen [17] and Ponce [11] independently obtained solutions on the relations between and n. They proved that for f ∈ H k ∩ W k; 1 suDciently small such that k ¿ n + 4 and n=2 ¿ , problem (1) – (2) admits a unique global small solution. In both papers, the method of decay rates of the L2 and L∞ norms of the solution were used. Later on, Zheng [16] ∗ Fax: +1-773-995-3767. E-mail address: [email protected] (T.A. Kwembe).
c 2002 Elsevier Science Ltd. All rights reserved. 0362-546X/02/$ - see front matter PII: S 0 3 6 2 - 5 4 6 X ( 0 1 ) 0 0 7 7 1 - 4
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used the method of global iteration combined with that of decay rates to re>ne the results of their earlier work. In this paper, though, I have employed the application of potential operators theory to study the weak solutions of problem (1) – (2) when the initial data f is in some Lp space. On using the techniques employed in [1–3,9,10], I have established the existence and uniqueness of maximal solutions when n=2 ¿ ( + 1)=; n ¿ 2 and the initial data is very small in the norm of de>nition. The paper also considers regularity results of maximal solutions and pointwise a.e. convergence of solutions in some Lp spaces, where p = n=2. Throughout the paper, x = (x1 ; : : : ; x n ) will denote a point in Rn : |x| = (x12 + · · · + 2 1=2 x n) will take the usual meaning of the distance of the point x from the origin. Lp (Rn ); 1 6 p ¡ ∞, will denote the space of Lebesgue measurable functions over Rn called the Lp space, and for all f ∈ Lp (Rn ), ||f||p will denote the Lp norm of the function f de>ned on Rn . We also de>ne u∗ (x) = supt¿0 |u(x; t)| to be the maximal function of u(x; t) over t ¿ 0 de>ned on Rn . In this case ||u∗ ||p = [ Rn (supt¿0 |u(x; t)|)p d x]1=p will denote the Lp norm of the function u∗ (x) on Rn . In order to establish the pointwise convergence result, we will consider the solution of the problem in the in>nite cylinder ST = Rn × [0; T ) for the initial data f ∈ Lp (Rn ); p =n=2, n=2 ¿ ( + 1)=. Under this consideration, we introduce the norm ||u∗ ||p (T ) = [ Rn (sup0¡t¡T |u(x; t)|)p d x]1=p ; p = n=2, n=2 ¿ (+1)=, where we now de>ne u∗ (x) = sup0¡t¡T |u(x; t)|. These norms have been introduced in [1,2,9]. If we let W (x; t) be the fundamental solution of the diKusion equation ut = 5u, then it is well known that a formal integral solution of (1) – (2) is given by t W (x − y; t − )u+1 (y; ) dy d + W (x − y; t)f(y) dy: (3) u(x; t) = 0
Rn
Rn
If we further let be the perturbed nonlinear operator de>ned by t W (x − y; t − )u+1 (y; ) dy d + W (x − y; t)f(y) dy: (4) (u)(x; t) = 0
Rn
Rn
Then by using the method of successive approximations we showed that the iterations uk+1 = (uk ) converges in the norm of de>nition to the solution of (3) and that the mapping is a contraction mapping a ball of radius say s0 into itself. Throughout this paper, we consider a solution u of (3) for all t ¿ 0 to be a weak solution of (1) – (2) whenever the integrals involved exist in the Lebesgue sense for all values of t ¿ 0 over Rn . The instrument for obtaining results in this paper is the Sobolev embedding Theorem 1 whose proof can be found in [15]. Theorem 1. Let 0 ¡ ¡ n; 1 6 p ¡ q ¡ ∞; 1=q = 1=p − =n. (a) If f ∈ Lp (Rn ); then (I f)(x) = [()]−1 Rn |x − y|−n f(y) dy; () =
[ 3=2 2 (=2)] (n=2 − =2)
converges absolutely for almost every x. (b) If; in addition; 1 ¡ p; then ||(I f)||q 6 AP; q ||f||p .
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In what follows, we shall consider the following standard Solonnikov estimates of the fundamental solution W which are given in [2,3,9,10,12,13]. |W (x; t; y; )| 6 c(|x − y| + |t − |1=2 )−n ;
(5)
where c ¿ 0 is a constant and n is the space dimension. The estimate for the spatial derivatives of W is given by |Dx" W (x; t; y; )| 6 C(|x − y| + |t − |1=2 )−n−|"| ;
(6)
where the derivatives exist in the distribution sense, and " = ("1 ; : : : ; "n ), |"| = "1 + · · · + "n , 9 9"1 9"2 9"n 9 9 " ; Dx = : ··· : · · · "n : Dx = 9x1 9x2 9x n 9x1"1 9x2"2 9x n We will also consider the Hardy–Littlewood Maximal function f∗ of a function f ∗ de>ned as f (x) = sup1=|Q| Q |f(y)| dy, where the supremum is taken over all cubes Q with center at x and edges parallel to the coordinate axes. Then it is proven in [15] that Theorem 2. Let 1 ¡ p 6 ∞ and f ∈ Lp (Rn ). Then f∗ ∈ Lp (Rn ) and ||f∗ ||p 6 C||f||p where C depends only on n and p. We shall refer to Theorem 2 as the maximal Theorem. We shall also consider the following known properties of W . (i) Rn W (x; t) d x = 1 (ii) supt¿0 Rn W (x − y; t)f(y) dy 6 cf∗ (x), where c is independent of f |"| t f(x−y) dy ∗ (iii) supt¿0 Rn |y| n+|"| +t n+|"| 6 Cf (x). (iv) W ∈ C0∞ . From property (iii), we have the well known result whose proof can be found in [15]. Theorem 3. Let H|"| (f)(x) = p
Rn
t |"| f(y) dy : |x − y|n+|"| + t n+|"|
n
If f ∈ L (R ); 1 6 p ¡ ∞; and |"| ¿ 0; then H|"| (f) ∈ Lp (Rn ) and ||H|"| (f)||p 6 C||f||p ; where C is a constant independent of f. 2. Estimates for the integral equation and xed-point properties Lemma 4. Let u(x; t) be a weak solution of problem (1) – (2) satisfying the integral equation (3). Then; the nonlinear transformation in (4) maps Ln=2 (Rn∗ ) to Ln=2 (Rn∗ ) and ||(u∗ )||n=2 6 C1 ||u∗ ||+1 n=2 + C2 ||f||n=2 ; n=2 ¿ + 1=; n ¿ 2; ¿ 0; where by n=2 n n=2 L (R∗ ) we mean the L norm of the maximal function u∗ (x) over Rn and the constants C1 ; C2 depends only on and n.
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t Proof. From (4) we have (u)(x; t) = 0 Rn W (x−y; t−)u+1 (y; ) dy d+ Rn W (x − y; t)f(y) dy. Then on utilizing estimate (5) we have t |u(y; )|+1 dy d |(u)(x; t)| 6 c + W (x − y; t)f(y) dy : 1=2 n Rn (|x − y| + |t − | ) Rn 0 On taking the supremum over t ¿ 0 of both sides of the inequality we have W (x − y; t)f(y) dy ; (u∗ )(x) 6 c C0 [u∗ (y)]+1 dy=|x − y|n−2 + sup Rn
t¿0
Rn
∞ √ where C0 is such that 0 ds=(1 + s)n 6 C0 . Then for p ¿ 1, we take the Lp norm of both sides and we use the Minkowski’s inequality on the right-hand side. For the ensuing >rst term on the right-hand side, we apply the Sobolev embedding Theorem 1. That is for p ¿ 1, we can >nd a q ¿ + 1 such that if, for 1=p = ( + 1)=q − 2=n; n ¿ 2 and p ¿ q= + 1 ¿ 1, [u∗ (x)]+1 ∈ Lq=+1 (Rn ); then [u∗ (y)]+1 dy ∗ +1 ∗ +1 n |x − y|n−2 6 C(q)||[u ] ||q=+1 = C(q)||u ||q : R P For the ensuing last term of the right-hand side of the inequality, property (ii) combined with Theorem 2 gives ||supt¿0 | Rn W (x − y; t)f(y) dy| ||p 6 C(p)||f||p . A combina+ C(P)||f||P . If we now require that tion of these gives ||(u∗ )||p 6 cC0 C(q)||u∗ ||+1 q p = q ¿ +1 in the equation 1=p = (+1)=q−(2=n), then we see that p = n=2 ¿ +1. From which we see that n=2 ¿ ( + 1)=. This concludes the proof of the lemma. Lemma 5. The nonlinear operator (u)(x; t) maps the ball {||u∗ ||n=2 6 s0 } into itself; if s0 is the smallest positive root of the equation s = C1 s+1 + C2 ||f||n=2 provided that C1 ; C2 and ||f||n=2 are chosen such that the graph of the line y = s remains underneath the graph of y = C1 s+1 + C2 ||f||n=2 ; and C2 ||f||n=2 6 s0 in 0 6 s 6 s0 . If C1 ( + 1)s1 ¡ 1; where 0 ¡ s1 ¡ s0 ; then is a contraction mapping in the ball of radius s0 . Proof. That maps the ball ||u∗ ||n=2 6 s0 into itself follows directly from Lemma 4. From Eq. (3) we have that u(x; t) = (u)(x; t). Therefore, u∗ (x) 6 (u∗ )(x). Hence for n=2 ¿ ( + 1)=; ¿ 0; we have by Lemma 4 that ||u∗ ||n=2 6 C1 ||u∗ ||+1 n=2 + C2 ||f||n=2 ; where Ci ; i = 1; 2 are constants depending only on and n. Let y = C1 s+1 +C2 ||f||n=2 . Then consider the function g(s) = C1 s+1 − s + C2 ||f||n=2 . By the Newton’s method of successive approximations, we can generate a succession of points s1 ; s2 ; : : : ; sk ; : : : 1 given by sk+1 = sk − g(sk )=g (sk ), k = 0; 1; 2; : : : ; sk = [C1 (+1)] 1= such that the sequence {sk } converges to a limit s0 ¿ 0, such that g(s0 ) = 0. Hence the function y intersect the line y = s at the point (s0 ; y(s0 )). Furthermore for C2 ||f||n=2 1, and small values of s, y ¡ s. Therefore, we can >nd a point say s1 with s0 ¡s1 such that a sequence of Newton’s approximation of g in the neighborhood of s1 converges to s1 and g(s1 ) = 0. Since y ¿ 0 in (s0 ; s1 ); y is convex in (s0 ; s1 ); and so in (0; s0 ); s 6 y and C2 ||f||n=2 6 y(s0 ). Hence, the line y = s remains underneath the graph of y = C1 s+1 +
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C2 ||f||n=2 and the graph of y = C1 s+1 + C2 ||f||n=2 remains underneath the line segment joining the points (0; C2 ||f||n=2 ) and (s0 ; s0 ). Thus y is convex in (0; s0 ) with respect to this line segment and hence we can >nd two points a, b in (0; s0 ) such that |y(b) − y(a)| 6 y ())|b − a| for some ) in (0; s0 ). Since g is a decreasing function on (0; s0 ), we have that g ¡ 0 for all s ∈ (0; s0 ). Thus, C1 ( + 1)s − 1 ¡ 0; s ∈ (0; s0 ). Thus, C1 ( + 1)s ¡ 1. Hence, the lemma follows if we replace s with ||u∗ ||n=2 and y with .
3. Main results We now state the main theorems of the paper. Theorem 6. Let u(x; t) be a weak solution of problem (1) – (2) satisfying the integral equation (3). Let f ∈ Ln=2 (Rn ); n=2 ¿ ( + 1)=; ¿ 0; and n ¿ 2 such that for a small *0 ¿ 0; ||f||n=2 ¡ *0 ; then; (i) problem (1) – (2) has at most one weak solution; and (ii) for all u(x; t) satisfying the integral equation (3); ||u∗ ||n=2 is 6nite. Proof. The proof follows from Lemmas 4 and 5. Theorem 7. Let u(x; t) be a weak solution of problem (1) – (2). If f ∈ Ln=2 (Rn ); n=2 ¿ ( + 1)=; ¿ 0; and n ¿ 2; then; (i) ||u∗ ||n=2 is 6nite and (ii) the function u(x; t) converges a.e. to f(x) in Ln=2 (Rn ) as t goes to zero. We will present the proof of Theorem 7 along the lines of the pointwise convergence argument in [1]. But, before we establish the proof of Theorem 7, we shall consider the following lemma: t Lemma 8. Let V (x; t) = 0 Rn W (x − y; t − )u+1 (y; ) dy d. If u ∈ Ln=2 (Rn × R+ ); n=2 ¿ ( + 1)=, ¿ 0. Then ||V ∗ ||n=2 (T ) 6 C||u∗ ||+1 n=2 (T ). Proof. On utilizing estimate (5) on V , we have t |u(y; )|+1 dy d |V (x; t)| 6 : 1=2 n 0 Rn (|x − y| + |t − | ) On taking the supremum over 0 ¡ t ¡ T on both sides we have sup0¡t¡T |V (x; t)| 6 T √ C0 Rn ([u∗ (y; T )]+1 dy)=(|x − y|n−2 ), where C0 is such that 0 ds=(1 + s)n 6 C0 . Then, if we assume that [u∗ (y; T )]+1 ∈ Lp=+1 (Rn ); we apply the Sobolev embedding Theorem for p ¿ + 1 such that 1=p = ( + 1)=p − 2=n; n ¿ 2, we get p ||sup0¡t¡T |V (x; t)|||p 6 C ||u∗ (T )+1 ||p=+1 = C(; n)||u∗ ||+1 n=2 (T ) +1 and the lemma follows. Furthermore, property (ii) and the maximal theorem tells us that Rn W (x−y; t)f(y) dy belongs to Ln=2 (Rn ) and ||sup0¡t¡T | Rn W (x−y; t)f(y) dy|
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||n=2 6 C||f||n=2 . Thus, the weak solution u of problem (1) – (2) belongs to the class of norms ||u∗ ||n=2 (T ). Proof of Theorem 7. By property (i), we have that u(x; t) − f(x) = V (x; t) + Rn W (x − y; t)[f(y) − f(x)] dy. On taking the Ln=2 (Rn ) norm of both sides, and then the limit as t approaches zero on both sides, we have the desired result after the following computation: By the continuity of f and a well known result in [15], we have that || Rn W (x − y; t)[f(y) − f(x)] dy||n=2 tends to zero with t. Also as a consequence of Lemma 8 and the fact that |V (x; t)| 6 C Rn ,(x − y; t)|x − y|−n+2 [u(y; T )]+1 dy, where t ,(y; t) = 0 |y|−2 (1 + |y|−1 s1=2 )−n ds, we see that V (x; t) tends to zero in Ln=2 norm as t tends to zero. Hence u(x; t) → f(x) a.e. in Ln=2 (Rn ) as t → 0.
4. Regularity results We shall employ the following standard result for the convolution derivative whose proof can be found in [15], to establish regularity results for the maximal solution of problem (1) – (2). Lemma 9. If 1 6 p ¡ ∞; f ∈ Lp (Rn ) and W ∈ C0∞ ; then f ∗ W ∈ C ∞ and D" (f ∗ W )(x) = (f ∗ D" W )(x). Theorem 10. Let |"| 6 M; be the order of the spatial distributional derivative of the maximal solution. Let u(x; t) be the weak solution of problem (1) – (2) given by (3). If n + |"|=2 ¿ + 1=; and f ∈ L((n+|"|))=2 (Rn ); then D" u(x; t) ∈ L((n+|"|))=2 (Rn∗ ) in the (+1) class of Lp (Rn∗ ) norm and ||D" u∗ ||(n+|"|)=2 6 C1 ||u∗ ||((n+|"|))=2 + C2 ||f||((n+|"|))=2 . t Proof. Let u(x; t) = 0 Rn W (x − y; t − )u+1 (y; ) dy d + Rn W (x − y; t)f(y) dy. t Then by Lemma 9, we have D" u(x; t) = 0 Rn (Dx" W (x − y; t − ))u+1 (y; ) dy d + Rn (Dx" W (x − y; t))f(y) dy. On using estimate (6), we have t |u(y; )|+1 dy d |f(y)| dy |D" u(x; t)| 6 C + C : 1=2 )n+|"| (|x − y| + |t − | (|x − y| + t 1=2 )n+|"| n n 0 R R Since (|x − y| + t 1=2 )n+|"| ¿ t −|"|=2 (|x − y|n+|"| + t (n+|"|)=2 ), we have that t |u(y; )|+1 dy d t |"|=2 |f(y)| dy |D" u(x; t)| 6 C + C : 1=2 n+|"| n+|"| + t (n+|"|)=2 0 Rn (|x − y| + |t − | ) Rn |x − y| Therefore, on taking the supremum of both sides over t ¿ 0, we have "
sup |D u(x; t)| 6 C0 C t¿0
Rn
[u∗ (y)]+1 dy + C sup |x − y|n+|"|−2 t¿0
Rn
t |"|=2 |f(y)| dy |x − y|n+|"| + t (n+|"|)=2
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∞ √ where C0 is such that 0 ds=(1 + s)n+|"| 6 C0 . Then we take the Lp (Rn ) norm of both sides using the Minkowski’s inequality on the right-hand side. Then we have two terms on the right-hand side to evaluate. Concerning the >rst term, if p, q are such that q ¿ + 1, and 1=p = + 1=q − 2=n + |"|; n + |"| ¿ 2, then we apply the Sobolev embedding theorem on the >rst term if [u∗ (y)]+1 ∈ Lq=+1 (Rn∗ ). We then apply property (iii) and the maximal Theorem 1 on the last term. Then, we have ||D" u∗ ||p 6 C0 CK(q=+1) ||u∗ ||+1 q +CK(p)||f||p . If we let p = q ¿ +1 in 1=p = ((+1)=q)−(2=(n+|"|)), then we have p = (n+|"|)=2 ¿ +1. From which we have that n+|"|=2 ¿ +1=. Hence, (+1) we have the required result of ||D" u∗ ||(n+|"|)=2 6 C1 ||u∗ ||(n+|"|)=2 + C2 ||f||(n+|"|)=2 ; n + |"|=2 ¿ + 1= and ¿ 0, if we take C1 = C0 CK((n + |"|)=2( + 1)), and C2 = CK((n + |"|)=2). Remark. The use of the above method for the case of classical solutions studied by some of the authors mentioned in the introduction may not hold. Especially when boundary value problems are introduced. For, if one consider the simple case (i) 9u=9t − 5u = u+1 ; (x; t) ∈ D × (0; ∞) ⊂ Rn × R+ ; (ii) u(x; 0) = f(x); x ∈ D, and (iii) 9u 9n = 0; (x; t)∈ 9D × (0; ∞), the existence of solution for all time t ¿ 0 depends on the sign of ( D f(x) d x) , ¿ 0. Since, if we let V (t) = D u(x; t) d x, and inte+1 grate (i) over D using (ii) and (iii), we have (iv) dV (t)=dt = D u (x; t) d x, and (v) V (0) = D u(x; 0) d x = D f(x) d x. On applying the Holder’s inequality of exponents ( + 1) and + 1=, we have that V (t) 6 |D| D u+1 (x; t) d x. Hence dV (t)=dt ¿ V (t)+1 =|D| , where |D| is the volume of the region D. Now, if we let y(t) be the solution of the Cauchy problem for the Ricatti equation (vi) dy(t)=dt = y(t)+1 =|D| , and (vii) y(0) = D f(x) d x ¿ 0. Then clearly we have that V (t) ¿ y(t). But, then f(x) d x D y(t) =
1= ; ¿ 0: 1 − =|D| D f(x) d x t If ( D f(x) d x) ¿ 0, then y(t) → ∞ as t → |D| =( D f(x) d x) . Hence in this case the solutions blows up at t0 = |D| =( D f(x) d x) . Hence, we can only obtain local solution on the interval [0; t0 ). Therefore, V (t) must blow up in a >nite time, and consequently so does the solution of (i) – (iii). However, if ( D f(x) d x) ¡ 0, then the solution exists for all time t ¿ 0. Thus, in this situation one has to be more careful t in establishing the aboveresults for the representative solution u(x; t) = 0 D W (x − y; t − )u+1 (y; ) dy d + D W (x − y; t)f(y) dy. An adjustment to maximal functions of u over an interval [0; t0 ) can be made if we de>ne u∗ (x) = sup06t¡t0 |u(x; t)| and then follow the treatment in establishing pointwise convergence. Then the existence and uniqueness as well as pointwise convergence results will hold.
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