Self-similar subsolutions and blow-up for nonlinear parabolic equations

NonlinearAmlysis.

Themy, Methods

Pergamon PII:

SO362-546X(97)00258-7

SELF-SIMILAR SUBSOLUTIONS NONLINEAR PARABOLIC PHILIPPE Laboratoire Institut

SOUPLET

AND BLOW-UP EQK4TIONS

and FRED

Analyse Geometric et Applications, Galilee, Universite Paris-Nord, 93430

email: soupletOmath.univ-parisl3.fr;

% Applications, Vol. 30, No. 7, pp. 46374641, 1997 Proc. 2nd World Congress of Nonlinear Amlysts 0 1997 Elsevier Science Ltd F’rinted in Great Britain. All rights reserved 0362-546X/97 $17.00 + 0.00

FOR

B. WEISSLER URA Villetaneuse,

CNRS

742 Prance

[email protected],fr

Abstract: For a wide class of nonlinear parabolic equations of the form uI -Aa=F(u,Vu), we give finite time blow-up results for large initial data. Blow-up time estimates are also provided. These results rely on a new method of comparison with suitable blowing up self-similar subsolutions. As a consequence, we improve several known results on reaction-diffusion equations, on generalized Burgers’ equations and on the equation of Chipot and Weissler. This method can also apply to degenerate equations of porous medium type, and provides a unified treatment for a large class of problems, both semilinear and quasilinear. Key tions,

words and gradient

phrases: term.

nonlinear

parabolic

equations,

finite

time

blow-up,

self-similar

subsolu-

1. INTRODUCTION

We consider

the following

parabolic

equation:

ut - Au = F(u,Vu),

t > 0, x E R,

with homogeneous Dirichlet boundary conditions and nonnegative initial St is a (possibly unbounded) regular domain in RN. Our main purpose is to give sufficient conditions for finite time blow-up general growth assumptions on F. The case when F(u, Vu) = f(u), e.g.:

(1.1) data under

4(z),

where

reasonably

ut - Au = up,

(1.2) b (or also f(u) = e”) h as een extensively studied over the past decades. Various sufficient conditions for blow-up have been provided and qualitative properties have also been investigated, such as nature of the blow-up set, rate and profile of blow-up, maximum existence time, and continuation after blow-up. By contrast, there has been a relatively small number of studies of blow-up for nonlinearities with a dependance on spatial derivatives of u. Convection-reaction-diffusion equations of generalized Burgers’ type ut - Au = up + uq-‘a.Vu

p, q > 1, a E IRN, a # 0, are considered prove blow-up for large function method, which term.

(1.3)

in [13, 1, 71. A mong other results, these authors initial data when p > q > 1. This is done essentially by the eigenapplies thanks to the particular conservative form of the gradient

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Another

example

in this

direction

Congress

of Nonlinear

Analysts

is the equation

ut - Au = up - ,u]VU]“,

(1.4)

p, q > 1, p > 0, introduced by Chipot and the second author [4] in order to investigate the possible effect of a damping gradient term on global existence or nonexistence. The first dynamics, where (1.4) describes the evolution of author [18] p ro p osed a model in population the population density of a biological species, under the effect of certain natural mechanisms. For this equation, the gradient term represents an absorption, which makes blow-up more difficult than in equation (1.3). In particular, the eigenfunction method does not apply, and blow-up is known to occur only in certain ranges of the parameters. The equation (1.4) has been studied in [4, 10, 5, 15, 2, 16-191. The existence of nonglobal solutions is unknown when (i) (ii) (iii)

2p/(p + 1) < q < p, q # 2, if N 2 2 or if p is large; q = 2p/(p + l), if p is large; 1 < q < 2p/(p + l), if N 2 3 and p 2 (N + 2)/(N - 2).

On the other the case: (iv)

hand,

global

q 2 p, R bounded

existence

for all nonnegative

initial

data

has been

proved

in

([5, 161).

(By slightly refining the argument of Fila [5], 1‘t is easily seen that this property remains valid for the convective equation (1.3).) Related to parabolic equations with gradient depending nonlinearity, let us mention the interesting question of gradient blow-up, that is ]Vu ] becoming unbounded at some point, as t approaches some finite time T, while u remains uniformly bounded. For instance, gradient blow-up has been exhibited for nonlinearities of the type

F(u,u,)

= f(~)Iu,I~-~u,,

772> 2 or

F(u,u,)

= eU*

(see [3, 6] and the references therein). It is known [ll] that gradient blow-up cannot occur if the growth of the nonlinearity is at most quadratic with respect to ]Vu], for instance in blow-up is excluded for equation the case of equation (1.3). On th e other hand, gradient (1.4), at least when G is bounded [16]. 2. BLOW-UP

In the work

[20], we consider

F(X,Y) F(X,Y)

RESULTS

the class of nonlinearities

F : IR

x

RN

> alXJ’, x 2 B, 0 5 IYI 5 x7, 2 -azIYIg, X 2 0, IYI 2 B,

--+ R, such that

(2.1)

for some p > q > 1, y > 1 and ul, a2, B > 0. (We also require some standard technical assumptions that guarantee the local existence and uniqueness and the validity of the comparison principle.) The growth conditions (2.1) are for instance fulfilled by any function of

Second World

the form F(u, Vu) than p, such as F(u,

Vu)

= up + G(u, Vu),

Congress of Nonlinear

w h ere G only contains

= up - ~u~]VU]~-’

F(u,

or

Vu)

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Analysts

power

terms

= up + UTpq-r-la.vU,

of total

degree

less

(2.2)

with p > q 2 1, 0 5 r < q. In particular, the nonlinearities in (1.3) or (1.4) for p > q 2 1 are allowed. It is also possible to handle more unusual nonlinearities, such as those satisfying

UP F(u,

Vu)

i 0 For such F, our main

result

if 0 5 ]Vu]

5 uy, 21 2 0 large,

= if u < 0 or ]Vu]

> l+

uT.

is the following.

THEOREM 1. Let $ E Wil”(0)

(s large enough), with 1,62 0, 4 f 0. (i) There exists some Xs > 0 (depending on $) such that for all X > Xs, the solution (1.1) with initial data 4 = Xll, blows-up in finite time in W1+ norm. (ii) There is some C > 0 such that

of

The basic idea of the proof is to compare u with a subsolution that blows up in finite time, In fact, we construct a self-similar subsolution, whose profile is compactly supported. Interestingly, whether or not (1.1) h as the invariance properties normally associated with self-similar solutions, we are able to find blowing-up self-similar subsolutions only assuming exponents will depend on these conditions, and the growth conditions (2.1). Th e similarity can be chosen within a certain range of values. Note that the existence of exact blow-up self-similar solutions of (1.4) has been proved [19], when q = 2p/(p + 1) (which is the only possible value for q). Under some additional mild restriction, e.g. for equations (1.3) and (1.4), the converse of estimate (ii) in Theorem 1 is also valid, under the form

hence determining the sharp asymptotics of the blow-up time. (This lower bound is easily obtained by comparison with a spatially constant solution of the form [(p - l)(T q]-ll(P-').) S'iml '1ar estimates were previously known ([12, 91) in the special case of equation (1.2), and were obtained by different techniques (eigenfunction method). As a corollary to Theorem 1, it follows that in any dimension N 2 1, and in any domain fl, the equation (1.4) admits bl ow-up solutions for all p > q > 1 and all p > 0. This, combined with (iv), establishes (for bounded domains) the conjecture set by Quittner [15]: The critical blow-up exponent for equation (1.4) is q = p

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(in the sense that blow-up can occur if and only if q < p). On this particular example, one can thus see that the growth conditions (2.1) in Theorem 1 are essentially optimal. Note that, from Theorem 1, one also recovers the above mentioned results on equation (1.3), and extends them to more general nonlinearities involving convection terms, such as in (2.2). Our method also applies to quasilinear parabolic equations of “porous medium” type: ut - A(]u]“--‘u)

= F(u,

Vu),

(2.3)

VU),

(2.4)

or of the type: Ut - IuI(J-l Au = F(u,

which were studied by several authors ([S, 14, 211) in the case F(u, Vu) = f(u). (More general elliptic operators can also be treated.) We prove that equations (2.3) and (2.4) do for sufficiently large data when p > cr 2 2. not admit global ( c 1assical) solutions In the case of unbounded domains, we obtain a criterion for blow-up in terms of the growth of 4 as (x] -+ co. Such type of results were first proved by Lee and Ni [12] in the different methods (using a variant of the case F(u, Vu) = up, with R = RN, by completely eigenfunction technique). We here extend this result to the general equation (1.1). Moreover, the growth condition at 00 need only be assumed in some smaller region, specifically a conic sector. In particular the domain Q need not be the whole space, but just contain some conic sector. In the case of equations (1.3) and (1.4), for instance, we obtain: 2. Assume that the domain R contains a conic sector R’ (or a half-line if N = 1). that 2p/(p + 1) < q < p (resp. q = (p + 1)/2) and let u be the solution of (1.4) ere exists a constant C = C(P) > 0 such that if $J E W,““(s2), 4 2 0, @eZies(1.3)1. Th THEOREM

Assume

,&,i;$*, then T’

lx1 2’(p-1)c#(x)

> c,

< 00.

To prove this result, we combine the technique of Theorem 1 with a resealing and translation argument, which allows one to “spread” the mass of the comparison function out to infinity. To conclude, let us stress that one of the important features of the method of self-similar subsolutions is its ability to recover a great part of the known results on the existence of blow-up, for semilinear and quasilinear equations of the form ut+Lu = F(u, Vu), and, at the same time, to improve them and give new results. Moreover, it allows a unified treatment for problems that previously had to be handled by different methods: eigenfunction method, energy method, other comparison methods, etc.

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of Nonlinear

Analysts

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[2] ALFONSI L. & WEISSLER F. B., Blow-up in R N for a parabolic equation with a damping nonlinear gradient term, Progress in nonlinear differential equations, Nonlinear diffusion equations and their equilibrium states, III, N. G. Lloyd et al. Editors, Birkhauser (1992). [3] ANGENENT S. & FILA M., Interior gradient blow up in a semilinear parabolic equation, Diff. Int. Equations, to appear. 141 CHIPOT M. & WEISSLER F. B.. Some blow uo results for a nonlinear parabolic problem with a ’ ’ gradient term, SIAM J. Math. Anal., ‘20, 4, 886-907-( 1989). [5] FILA M., Remarks on blow up for a nonlinear parabolic equation with a gradient term, Proc. Amer. Math. Sot., 111, 2, 795-801 (1991). [6] FILA M. & LIEBERMAN G. M., D erivative blowup and beyond for quasilinear parabolic equations, Diff. Int. Equations, 7, 3, 811-821 (1994). [7] FRIEDMAN A., Blow up of solutions of parabolic equations, Nonlinear diffusion equations and their equilibrium states, I, W. M. Ni et al. Editors, Springer (1988). [S] GALAKTIONOV V. A., A boundary value problem for the nonlinear parabolic equation ut-Aubt’= up, Differential’nye Uravneniya, 17, 836-842 (1981). [Russian] [9] GUI C. & WANG X., Life span of solutions of the Cauchy problem for a semilinear heat equation, J. Diff. Equations, 115, 166-172 (1995). [lo] KAWOHL B. & PELETIER L. A., Observations on blow up and dead cores for nonlinear parabolic equations, Math. Z., 202, 207-217 (1989). [ll] LADYZENSKAJA 0. A., SOLONNIKOV V. A. & URALCEVA N. N., Linear and quasilinear equations of parabolic type, Translations of Mathematical Monographs, A.M.S., Providence (1968). [12] LEE T. & NI W., Global existence, large time behaviour and life span of solutions of a semilinear parabolic Cauchy problem, Trans. Amer. Math. Sot., 333, 1, 365-378 (1992). [13] LEVINE H. A., PAYNE L. N., SACKS P. E. & STRAUGHAN B., Analysisofaconvectivereactiondiffusion equation (II), SIAM J. Math. Anal., 20, 1, 133-147 (1989). [14] LEVINE H. A. & SACKS P. E., S ome existence and nonexistence theorems for solutions of degenerate parabolic equations, J. Diff. Equations, 52, 135-161 (1984). [15] QUITTNER P., Blow-up for semilinear parabolic equations with a gradient term, Math. Meth. Appl. Sciences, 14, 413-417 (1991). [16] QUITTNER P., On global existence and stationary solutions for two classes of semilinear parabolic equations, Comment. Math. Univ. Carolinae, 34, 1, 105-124 (1993). [17] SOUPLET P., RBsultats d’explosion en temps fini pour une kquation de la chaleur non linbaire, C. R. Acad. SC. Paris, 321, SCrie I, 721-726 (1995). [18] SOUPLET P., Finite time blow up for a nonlinear parabolic equation with a gradient term and applications, Math. Meth. Appl. Sciences, in press. [19] SOUPLET P., TAYACHI S. & WEISSLER F. B., Exact self-similar blow-up of solutions of a semilinear parabolic equation with a nonlinear gradient term, Indiana Univ. Math. J., to appear. and blow-up for nonlinear parabolic [20] SOUPLET P. & WEISSLER F. B., S e If- semi . 1ar subsolutions equations, Technical report 96-10, UniversitC Paris-Nord, 15 p. (1996). [21] WIEGNER M., Blow-up for solutions of some degenerate parabolic equations, Diff. ht. Equations, 7, 6, 1641-1647 (1994).