Various behaviors of solutions for a semilinear heat equation after blowup

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1 Journal of Functional Analysis ( Various behaviors of solutions for a semilinear heat equation after blowup Noriko Mizoguchi Department of Mathematics, Tokyo Gakugei University, Nukuikita 4--, Koganei-Shi, Tokyo-to 84-85, Japan Received 9 September 3; received in revised form April 4; accepted 5 July 4 Communicated by M. Brezis Abstract The present paper is concerned with a Cauchy problem for a semilinear heat equation u t = Δu + u p in R N (,, u(x, = u (x in R N. (P We show that if p> N N N 4 N and N, then there exists a solution u i (i =,, 3 of (P which blows up at t = T i < +, becomes a regular solution for all t>t i and behaves as follows: (i lim t u (t =, (ii < lim inf t u (t lim sup t u (t < +, (iii lim t u 3 (t =+, where denotes the supremum norm in R N. 4 Elsevier Inc. All rights reserved. MSC: 35K; 35K55; 58K57 Keywords: Incomplete blowup; Peaking solution; Semilinear heat equation Fax: address: (N. Mizoguchi. -36/$ - see front matter 4 Elsevier Inc. All rights reserved. doi:.6/j.jfa.4.7.4

2 . Introduction N. Mizoguchi / Journal of Functional Analysis ( This paper is concerned with a Cauchy problem for a semilinear heat equation { ut = Δu + u p in R N (,, u(x, = u (x inr N (. with p> and u L (R N. We are interested in the continuation of a solution of (. which blows up in finite time. Here, a solution u is said to blow up at t = T<+ if lim sup t T u(t = + with the supremum norm in R N. When a solution u of (. blows up at t = T, the blowup is called complete if the proper continuation for t>t identically equals + in R N (T,, and incomplete otherwise. We refer to Section of Galaktionov and Vazquez [5] for the definition of proper solution and its important properties. In the case of (N p < N +, only the complete blowup is possible by Baras and Cohen []. On the other hand, Ni et al. [4] obtained a global L -solution of the Cauchy Dirichlet problem for the equation in (. in a convex domain which is unbounded in L when p (N + /(N and N 3. Let { + if N, p L = + N 6 if N. (. We suppose that (N + /(N <p<p L and N 3. Then any radially symmetric solution in a ball obtained by [4] blows up in finite time by [5]. The author obtained a radially symmetric global L -solution of (. with initial data compactly supported which exhibits incomplete blowup in finite time in []. However, there is no more detailed information on the behaviors of these solutions after blowup time. It was shown in [5] that there exists a solution u of (. such that u(x, t = (T t /(p f ((T t / x in R N [,T with a spatially inhomogeneous positive regular solution f(r for f rr + N f r r r f r p f + f p = in (, (.3 and u(x, t = (t T /(p g((t T / x in R N (T, with a spatially inhomogeneous positive regular solution g(r for g rr + N g r + r r g r + p g + gp = in (,. (.4 This solution is a backward self-similar blowup solution for t [,T and a forward self-similar solution for t (T, +. We note that if (N + /(N <p<p L and

3 6 N. Mizoguchi / Journal of Functional Analysis (5 4 7 N 3, then there is a positive regular solution f(r for (.3 such that f(r= cr p + o(r p as r with some c<c by Budd and Qi [] and Lepin [8,9] (see [5], where c = { ( N } p p p (.5 so that v (r c r /(p is the radially symmetric singular steady state of (.. However, there seems to be no spatially inhomogeneous positive regular solution of (.3 for p > p L. In fact, Plecháč and Šverák [5] got a numerical result which suggests it. The author gave a rigorous proof when p> + 7/(N and N in []. Consequently, we cannot ect to construct a peaking solution by connecting a backward self-similar blowup solution with a forward self-similar solution as done in [5]. Here, peaking solution means a solution which blows up at some T < + and becomes a regular solution for all t>t. On the other hand, for any c (,c there exists a positive regular solution g(r for (.4 with p>(n+ /(N and N 3 such that g(r = cr p + o (r p as r (.6 by [5]. Furthermore, Souplet and Weissler [7] obtained a positive regular solution g(r for (.4 satisfying (.6 for each c (,c + δ with sufficiently small δ > in the case of (N + /(N <p<p JL and N 3, where { + if N, p JL = N N N 4 N (< p if N. (.7 L When a radially symmetric global L -solution u of the Cauchy Dirichlet problem for the equation in (. in a ball with (N + /(N <p<p JL blows up in finite time, the continuation as proper solution is a global regular solution after the blowup time if (u t changes its sign exactly once by Fila et al. [4]. We note that peaking solutions of (. given in the previous results converge to zero as t. Our purpose of the present paper is to prove the following. Theorem.. If p>p JL, then there exists a proper solution u i (i =,, 3 of (. which blows up at t = T i for some T i < +, becomes a regular solution for all t>t i and behaves as follows: (i lim t u (t =, (ii < lim inf t u (t lim sup t u (t < +, (iii lim t u 3 (t =+.

4 N. Mizoguchi / Journal of Functional Analysis ( This paper is organized as follows: In Section, we first get a result on the behavior of solutions for (. near the spatial infinity. We also consider (. with singular initial data, which is related to the regularization of a solution after blowup. Section 3 is devoted to the proof of Theorem.. In order to obtain a peaking solution, we need a result in [3] which is an analogue of the result in Herrero and Velázquez [7]. A recent result due to Poláčik and Yanagida [6] is applied to decide the behavior of the peaking solution as t.. Preliminary results We begin this section by the following lemma. Denote by B r the ball with radius r centered at the origin in R N. Lemma.. Let p > N/(N and T >. Suppose that u is a solution of (. with u L satisfies u(x, t < v ( x in (R N \B r [,T and u(x, t c x p in (R N \{} [,T (. for some c, r >. If there are k, a,r > with sufficiently large R such that v ( x u (x a x k in R N \B R, then it holds v ( x u(x, t a x k in (R N \B R [,T] for some a > and R R. Proof. We first write u as x y u(x, t= (4πt N/ u (y R N 4t t + (4π(t s N/ u(y, s p x y ds. (. R N 4(t s

5 8 N. Mizoguchi / Journal of Functional Analysis (5 4 7 Since v is the unique solution of (. with initial data v for p>p JL by [5], we get x y v ( x = (4πt N/ v ( y R N 4t t + (4π(t s N/ v ( y p x y ( R N 4(t s for x =. Putting ũ(x, t = v ( x u(x, t and we see g(x, t = v ( x p u(x, t p v ( x u(x, t, dy ds (.3 g(x, t pc p x in (R N \{} [,T (.4 with c = max{c,c}. Then it follows from (., (.3 that x y ũ(x, t= (4πt N/ ũ(y, R N 4t t + (4π(t s N/ g(y,sũ(y, s R N for x =. Setting I = ( x y 4(t s x y (4πt N/ ũ(y, R N 4t dy ds and we have I = t x y (4π(t s N/ g(y,sũ(y, s ds, R N 4(t s ũ(x, t = I + I. (.5 Put ũ + (x = max{ũ(x,, } and ũ (x = max{ ũ(x,, }

6 N. Mizoguchi / Journal of Functional Analysis ( and let U ± be the solution of the heat equation in R N with U ± (x, = ũ ± (x. Then U ± is represented as U ± (x, t = x y (4πt N/ ũ ± (y. (.6 R N 4t Take R>R sufficiently large, and let x R and t (,T. Setting I = x y R x y ũ + (y 4t and I = x y > R x y ũ + (y, 4t we have It is immediate that Since x y ũ + (y = I + I. (.7 R N 4t ( I ũ + 8t ( R = ũ + (8πt N/ ( R 3t x y R N 8t. (.8 ũ + (x a x k for x with x R by the assumption, we get I a ( 3R =a ( 3R k x y x y R 4t k { x y R N 4t

7 N. Mizoguchi / Journal of Functional Analysis (5 4 7 a ( 3R x y > R } x y 4t k { } (4πt N/ (8πt N/ ( R 3t (.9 in the same way as in (.8. It follows from (.6 (.9 that U + (x, t C R k C ( R 3t (. for some C,C >. Similar to (.8, there is C 3 > such that Hence it holds U (x, t C 3 ( R. (. 3t I = U + (x, t U (x, t C 4 R k (. for some C 4 > ifr> is sufficiently large. Since ũ(x, t > in(r N \R r [,T by the hypothesis, it follows from (., (.4 that for x R >r and t (,T x y g(y,sũ(y, s R N 4(t s pc p c ( (R r 4(t s pc p cr N p ( (R r 4(t s y p dy y r and hence x y (4π(t s N/ g(y,sũ(y, s R N 4(t s pc p cr N p ( (R r N/ (4π(t s 4(t s pc p cr N p ( (R r ( R. N/ (4π(t s 8(t s 3T

8 N. Mizoguchi / Journal of Functional Analysis (5 4 7 Consequently there exists C 5 > such that I C 5 ( R 3T (.3 since (4π(t s N/ ( (R r C 6 8(t s for s, t with <s<t<t with some C 6 >. The inequalities (.5, (., (.3 imply ũ(x, t C 7 R k C 7 x k in (R N \B R [,T for some C 7 >. We next consider (. with singular initial data. Lemma.. Let p>p JL. If u is radially symmetric and satisfies u ( x v ( x and u ( x / v ( x in R N \{}, then the proper solution u t>. of (. with initial data u fulfills u(t L (R N for all Proof. Since u(r, t v (r in (, [, with r = x, weget u t u rr + N r Putting v(r, t = r /(p u(r, t, it holds u r + cp r u in (, (,. v t v rr + N 4 p r v r in (, (,. (.4

9 N. Mizoguchi / Journal of Functional Analysis (5 4 7 Let V be the solution of { V t = V rr + N 4 p V(r, = v(r, r V r in (, (,, in [,. Since v(r, c and v(r, / c in [,, we see V(r,t<c in [, (, and hence it follows from (.4 that v(r,t<c in [, (,. For any r,t >, let { c = max v(r, t : r r, t } t t. Denote by ũ(r, t a forward self-similar solution with initial data c r /(p with c < c <c, whose existence is mentioned in Introduction. Then there is t (,t / such that ( u r, t + t ũ(r, t in [,r ] [,t ] and hence ( u r, t + t ũ(r, t in [,r ]. (.5 Since u(r, t + t / v (r in [r,, it follows from (.5 that u(t + t / < +. It is shown in [6] that if initial data U L satisfies U (x v ( x for all x R N \{}, then the solution of (. with initial data U exists globally in time in the classical sense. Therefore, we see u(t < +. Since t > is arbitrary, this completes the proof.

10 3. Proof of the main result N. Mizoguchi / Journal of Functional Analysis ( In the present section, we prove Theorem.. To do that, we need results in [6]. Let v m be the solution of { v + N r v + v p = in(,, v( = m>, v ( =. It was given in Li [] that if p>p JL, then v m is increasing with respect to m (3. and v m (r = v (r k(mr α + o(r α as r (3. for some k(m >, where α = (N + β 4(p c p (3.3 with β = N 4/(p. They allowed solutions to change signs in [6]. However, this paper is concerned with only nonnegative solutions, so we refer to their result in the following manner necessary to our situation. Lemma 3.. Let p>p JL and suppose that u (x v ( x in R N \{}. Then a solution u of (. with initial data u L exists globally in time in the classical sense and (i if lim x x /(p u (x =, then lim t u(t = ; (ii if lim x x α u (x v m ( x = for a solution v m of (3., then lim t u(t v m = ; (iii if lim x x α u (x v ( x =, then lim t u(t =+, where α is as in (3.3. with Putting w(y,s = (T t /(p u(x, t y = (T t / x and s = log(t t for a solution u of (. and T>, w satisfies { w s = Δw y w p w + wp in R N ( log T,, w(y, log T= T /(p u (T / y in R N.

11 4 N. Mizoguchi / Journal of Functional Analysis (5 4 7 This is represented as { w s = w rr + N r w r r w r p w + wp in (, ( log T,, w(r, log T= T /(p u (T / r in [,. (3.4 with r = y in the radially symmetric case. It is immediate that v is also a singular steady state of (3.4. Let { } L w = h L loc : h(r r N ( r < + 4 and H w ={h H loc : h, h L w }. Denote by,w the natural norm in L w. Define an operator A by Aφ = φ N φ + r r φ + p φ pcp r φ. It was shown originally in [6,7] and also in [] that if p>p JL, then the spectrum of A consists of countable eigenvalues {λ j } such that λ j = α + + j for j =,,,... p with α in (3.3 and its corresponding eigenfunction φ j is given by φ j (r = c j r α M ( j,α + N r ; 4 with the standard Kummer function M(a,b; ξ and c j > taken so that φ j,w =. Then it holds φ j (r = c j r λ j p + o(r λ j p as r for some constant c j and φ j (r = c j r α + o(r α as r. (3.5 The following result was given in [3] which was based on the method in [7].

12 N. Mizoguchi / Journal of Functional Analysis ( Lemma 3.. If p>p JL, then for any even integer l with λ l > there exists a solution u i (i =,, 3 of (. with radially symmetric initial data u i, L which blows up at t = T i for some T i < + and satisfies (i u i, has l intersections with v ; ( (ii- u, (r ar p +δ for sufficiently large r with some a,δ > ; (ii- u, (r v (r br α for sufficiently large r with some b> and u, (r v m (r for all r, where v m is a solution of (3. with k(m>b in (3.; (ii-3 u 3, (r v (r cr ( α +δ3 for sufficiently large r with some c, δ 3 > and u 3, (r u (r for all r, where u is the initial data of a global classical solution u of (. such that u(t as t ; (iii Let ( η = λ l α, γ = η p p, (3.6 c l be the constant in (3.5 with j = l and v i be the solution of (3. with k(m = k i in (3. for k i with k >c l >k, for i =,. For a solution w of (3.4 associated with u, e γs v (e ηs r<w(r,s<e γs v (e ηs r for r [,Ke ηs ] and s log T with some K>; (iv For sufficiently small ε>, the solution w satisfies ( w(r, s v (r + e λls φ l (r εe λ ls r α + r λ l p for r [Ke ηs,e σs ] and s [ log T, with some σ >. We are now in a position to prove Theorem.. Proof of Theorem.. Let u i be the solution with blowup time T i obtained in Lemma 3. for i =,, 3. Putting u i (r, T i = lim t Ti u i (r, t for i =,, 3, it follows from (i, (iv that u i (r, T i v (r in (, since the number of intersections between u i and v is nonincreasing with respect to t by Chen and Poláčik [3]. By Lemma., there are a,b,c,r > with sufficiently large r such that for each r r (i u (r, T a r ( p +δ ;

13 6 N. Mizoguchi / Journal of Functional Analysis (5 4 7 (ii v m (r u (r, T v (r b r α ; (iii u(r, T 3 u 3 (r, T 3 v (r c r ( α +δ3. Let also denote by u i the proper solution of (. for i =,, 3. Put u i,n (r, = min{u i (r, T i, n} for positive integer n and let u i,n be the solution of (. with initial data u i,n (r,. Then there are a,b,c,r > such that for small δ > the above (i (iii with u i (r, T i, a,b,c,r replaced by u i,n (r, δ, a,b,c,r are valid for all n by Lemma.. Therefore u i (r, T i + δ satisfies the same inequality for i =,, 3 passing to the limit as n. According to Lemma., u i becomes a regular solution for all t>t i for i =,, 3. Applying Lemma 3. to u i (r, T i + δ for i =,, 3, we complete the proof. Acknowledgments The author resses her gratitude to Professor Marek Fila for giving me useful information. She is also grateful to the referee for helpful suggestions. References [] P. Baras, L. Cohen, Complete blow-up after T max for the solution of a semilinear heat equation, J. Funct. Anal. 7 ( [] C.J. Budd, Y.-W. Qi, The existence of bounded solutions of a semilinear elliptic equation, J. Differential Equations 8 ( [3] X.-Y. Chen, P. Poláčik, Asymptotic periodicity of positive solutions of reaction diffusion equations on a ball, J. Reine Angew. Math. 47 ( [4] M. Fila, H. Matano, P. Poláčik, Immediate regularization after blow-up, Proceedings of Equadiff 3. [5] V.A. Galaktionov, J.L. Vazquez, Continuation of blowup solutions of nonlinear heat equations in several dimensions, Comm. Pure Appl. Math. 5 ( [6] M.A. Herrero, J.J.L. Velázquez, Explosion de solutions des équations paraboliques semilinéaires supercritiques, C. R. Acad. Sci. Paris 39 ( [7] M.A. Herrero, J.J.L. Velázquez, A blow up result for semilinear heat equations in the supercritical case, preprint. [8] L.A. Lepin, Countable spectrum of the eigenfunctions of the nonlinear heat equation with distributed parameters, Differentsial nye Uravneniya 4 ( (Engl. transl.: Differential Equations 4 ( [9] L.A. Lepin, Self-similar solutions of a semilinear heat equation, Mat. Model ( (in Russian. [] Y. Li, Asymptotic behavior of positive solutions of equation Δu+K(xu p =inr N, J. Differential Equations 95 ( [] N. Mizoguchi, On the behavior of solutions for a semilinear parabolic equation with supercritical nonlinearity, Math. Z. 39 ( 5 9. [] N. Mizoguchi, Blowup behavior of solutions for a semilinear heat equation with supercritical nonlinearity, J. Differential Equations, in press. [3] N. Mizoguchi, Type II blowup for a semilinear heat equation, in press. [4] W.-M. Ni, P.E. Sacks, J. Tavantzis, On the asymptotic behavior of solutions of certain quasi-linear parabolic equations, J. Differential Equations 54 ( [5] P. Plecháč, V. Šverák, Singular and regular solutions of a non-linear parabolic system, preprint.

14 N. Mizoguchi / Journal of Functional Analysis ( [6] P. Poláčik, E. Yanagida, On bounded and unbounded global solutions of a supercritical semilinear heat equation, Math. Ann. 37 ( [7] P. Souplet, F.B. Weissler, Regular self-similar solutions of the nonlinear heat equation with initial data above the singular steady state, Ann. Inst. H. Poincaré (

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