Simultaneous vs. non simultaneous blow-up

Transcription

1 Simultaneous vs. non simultaneous blow-up Juan Pablo Pinasco and Julio D. Rossi Departamento de Matemática, F..E y N., UBA (428) Buenos Aires, Argentina. Abstract In this paper we study the possibility of simultaneous blow up for positive solutions of a system of two heat equations, u t = u, v t = v, in a bounded smooth domain Ω, with boundary conditions u η η = u p v p 2, = up 2 v p 22. We prove that if u has blow up then v must blow up if and only if p > and p 2 > p. Introduction. In this note we study blowing up solutions of the following parabolic system, { ut = u in Ω (0, T ), (.) v t = v in Ω (0, T ), with boundary conditions, { u η = up v p 2 on (0, T ), η = up2 v p22 on (0, T ), (.2) and initial data, { u(x, 0) = u0 (x) in Ω, v(x, 0) = v 0 (x) in Ω. (.3) Throughout this paper we assume that p ij 0 and u 0, v 0 > 0. Under the hypothesis, p >, p 22 > or (p )(p 22 ) p 2 p 2 < 0 it is proved in [6] (see also [7]) that the solution (u, v) blows up in finite time, T. At that time T we have, by the result of [6], lim sup t T u(, t) L (Ω) + v(, t) L (Ω) = +. Supported by Universidad de Buenos Aires under grantex046 and ONIET (Argentina) AMS-Subj.class : 35B40, 35J65, 35K60. Keywords : blow-up, parabolic systems, nonlinear boundary conditions.

2 We observe that, a priori, there is no reason that guarantee that both functions, u and v, go to infinity simultaneously at time T. In this work we address this problem and characterie the simultaneous blow-up in the following sense: suppose that u goes to infinity at time T then v may remain bounded up to this time T if and only if p > (this guarantee the blow up for u) and p 2 < p (this implies that the coupling between u and v is weak ). We prove: Theorem. Assume that u blows up at time T and that v remains bounded up to time T, then p > and p 2 < p. Theorem.2 If p > and p 2 < p then there exists initial data u 0, v 0 such that u has blow up but v remains bounded. To prove this theorems we have to assume that the blow up rate for a single equation w t = w in Ω (0, T ), w η cwq on (0, T ), (.4) w(x, 0) = w 0 (x) in Ω, with q >, is given by max w(, t) Ω (T t) 2(q ) (.5) This estimate was proved in [2] and [3] under the hypothesis q < N 2 (for N =, 2 this requirement is not necessary) and some assumptions on the initial data. 2 Proof of the main results We begin by Theorem.. Assume that (u, v) has finite time blow up T and that v remains bounded up to time T. Therefore, u must blow-up at time T. As u is a solution of u t = u in Ω (0, T ), u η = up v p 2 u p on (0, T ), w(x, 0) = w 0 (x) in Ω, N (2.) that has finite time blow up, we must have p > (see [8]). Now we want to prove that p 2 < p. Suppose not and let Γ(x, t) be the fundamental solution of the heat equation, namely Γ(x, t) = exp (4πt) n/2 ( x 2 4t ). 2

3 Now for x, using Green s identity and the jump relation (see []) we have v(x, t) = Γ(x y, t )v(y, ) dy + 2 Ω t t Γ η (x y, t τ)v(y, τ) ds ydτ. η (y, τ)γ(x y, t τ) ds ydτ (2.2) Now we set V (t) = sup Ω [,t] v. Since Ω is smooth, for instance +α, Γ satisfies (see []) Γ (x y, t τ) η (t τ) µ x y n+ 2µ α As η = up 2 v p 22 u p 2, 2 V (t) t u p 2 (y, τ)γ(x y, t τ) ds y dτ V (t)(t ) µ. Now, we choose such that (T ) µ < /2 then V (t) t u p2 (y, τ)γ(x y, t τ) ds y dτ From [5] we have that the blow up set for u is contained in, so let x 0 be a blow up point for u. It is proved in [3] that under the hypothesis (.5) the blow up limit is nontrivial, that is lim inf t T inf u(x 0 + x 2(p T t, t)(t t) ) 0 x K Then there exists a constant c such that u(x 0 + x T t, t) c (T t) 2(p ) x K With this bound for u we obtain, t V (t) (T τ) p 2 2(p ) t { y x 0 K(T t) /2 } p 2 dτ. 2(p (T τ) ) (t τ) /2 Γ(x y, t τ) ds y dτ One can check that the integral in the right hand side diverges as t T if, and hence v must blow up at time T, a contradiction. p 2 2(p ) 2 3

4 Now we prove Theorem.2. We want to choose u 0, v 0 in order to obtain a blowing up solution (u, v) with v bounded. First assume that p 22 >. We choose 0 < v 0 small and observe that, as p >, every positive solution has finite time blow up and u verifies (.4), so, by assumption, we have u(x, t) (T τ) 2(p ) From this we observe that if u 0 is large with v 0 fixed then T becomes small. As before, we want to use the representation formula obtained from the fundamental solution. As η = up2 v p22 v p22 (T t) p2 2(p from (2.2) we obtain, t V (t) V () + V (t)p22 p 2 2 dτ + V (t)(t ) µ. 2(p (T τ) ) (t τ) /2 We choose u 0 large enough in order to get T small such that (T ) µ < /8. p Since 2 2(p ) < /2, the integral converges, and we can assume that it is less than /8 if u 0 is large. As long as V (t) is less that one V (t) p 22 < V (t) and hence we get, V (t) V (). 8 We can conclude that v remains bounded up to time T. Next, assume that p 22, we choose v 0 > in order to obtain V (t) > and with the same arguments as before we obtain, Now V (t) p 22 4 V (t) 8 V (t)p 22 V (). < V (t) and then, V (t) V (). 8 The right hand side of the last inequality is bounded uniformly in t as we wanted to prove.. References [] A. Friedman, Partial Differential Equations of Parabolic Type, Prentice- Hall, Englewood liffs, NJ (964). [2] B. Hu and H. M. Yin, The Profile Near Blow-up Time for the Solution of the Heat Equation with a Nonlinear Boundary ondition, Trans. Amer. Math. Soc. vol 346 () (995),

5 [3] B. Hu, Nonexistence of a positive solution of the Laplace equation with a nonlinear boundary condition, Diff. and Int. Eq., V. 7, Nro. 2, (994), [4] B. Hu, Nondegeneracy and single-point-blowup for solution of the heat equation with as nonlinear boundary condition, J. Math. Sci. Univ. Tokyo, V., (994), [5] D. F. Rial and J. D. Rossi, Blow-up results and localiation of blow-up points in an n-dimensional smooth domain. Duke Math. Jour. Vol. 88 (2), (997), [6] J. D. Rossi, On existence and nonexistence in the large for an N- dimensional system of heat equations whit nontrivial coupling at the boundary. New Zealand Jour. of Math. Vol 26, (997), [7] J. D. Rossi and N. Wolanski, Global Existence and Nonexistence for a Parabolic System with Nonlinear Boundary onditions, to appear in Diff. Int. Eq. [8] W. Walter On Existence and Nonexistence in the Large of Solutions of Parabolic Differential Equations with a Nonlinear Boundary ondition, SIAM J. Math. Anal. Vol 6(), (975),