The question of blow-up of solutions to nonlinear parabolic equations and systems has received considerable attention in the recent literature. We refer to the book of Quittner-Souplet {QS} , as well as to the survey paper of Bandle and Bruner {BB} and the paper of Vazquez{V}(More recent results in{PPV} and the references therein). In practical situations one would like to know among other things whether the solution blows up and, if so, at which time $t^*$ blow-up occurs. When the solution does blow up at some finite $t^*$, this time can seldom be determined explicitly, so much effort has been devoted to the calculation of bounds for $t^*$. Most of the methods used until recently have yielded only upper bounds for $t^*$, so that in particular problems in which blow-up has to be avoided, they are of little value. To the authors' knowledge some of the first work on lower bounds for $t^*$ was by Weissler {W1}, {W2}, but during the past three or four years a number of papers deriving lower bounds for $t^*$ in various problems have appeared, beginning with the paper of Payne and Schaefer {PS1}. This paper deals with the blow-up of the solution to the following semilinear second order parabolic equation with nonlinear boundary conditions: $u_t = \nabla ( |\nabla u|^{2p} \nabla u), x in Omega , t in (0,t^*),$ $|\nabla u|^{2p} \frac {\partial u}{\partial n}= f(u), x in \partial \Omega,t in (0,t^*)$, $u (x,0) = u_0( x)\geq 0, x in \Omega.$ p is some nonnegative parameter, n stands for the outward normal. It is shown that under certain conditions on the nonlinearities and data, blow-up will occur at some finite time. When blow-up does occur an lower bound for the blow-up time is obtained, by using a Sobolev-type inequality. An upper bound is also derived. References {BB} C.Bandle, H. Bruner, Blow-up in diffusion equations. A survey, J.Comput. Appl. Math. 97 (1998) 3--22. {PPV} L.E.Payne, G.A.Philippin, S. Vernier-Piro, Blow- up phenomena for a semilinear heat equation with nonlinear boundary condition I, Z. Angew. Math. Phys., 61 (2010) 999-1007. {PS1} L.E.Payne, P.W. Schaefer, Lower bound for blow-up time in parabolic problems under Neumann conditions, Applicable Analysis 85 (2006) 1301--1311. {QS} R Quittner and P. Souplet, Superlinear parabolic problems. Blow-up, global existence and steady states, Birkh\"auser Advanced Texts, Basel, (2007) {V} J. L. Vazquez, The problems of blow-up for nonlinear heat equations. Complete blow-up and avalanche formation, Rend. Mat. Acc. Lincei s. 9 15 (2004) 281--300. {W1} F.B.Weissler, Local existence and nonexistence for semilinear parabolic equations in $L^p$, Indiana Univ. Math. J. 29 (1980) 79--102 . {W2} F.B.Weissler, Existence and nonexistence of global solutions for a heat equation, Israel J.Math. 38 (1981) n.1-2, 29--40.
Blow-up phenomena for a semilinear heat equation with nonlinear boundary condition, II
PIRO, STELLA
2010-01-01
Abstract
The question of blow-up of solutions to nonlinear parabolic equations and systems has received considerable attention in the recent literature. We refer to the book of Quittner-Souplet {QS} , as well as to the survey paper of Bandle and Bruner {BB} and the paper of Vazquez{V}(More recent results in{PPV} and the references therein). In practical situations one would like to know among other things whether the solution blows up and, if so, at which time $t^*$ blow-up occurs. When the solution does blow up at some finite $t^*$, this time can seldom be determined explicitly, so much effort has been devoted to the calculation of bounds for $t^*$. Most of the methods used until recently have yielded only upper bounds for $t^*$, so that in particular problems in which blow-up has to be avoided, they are of little value. To the authors' knowledge some of the first work on lower bounds for $t^*$ was by Weissler {W1}, {W2}, but during the past three or four years a number of papers deriving lower bounds for $t^*$ in various problems have appeared, beginning with the paper of Payne and Schaefer {PS1}. This paper deals with the blow-up of the solution to the following semilinear second order parabolic equation with nonlinear boundary conditions: $u_t = \nabla ( |\nabla u|^{2p} \nabla u), x in Omega , t in (0,t^*),$ $|\nabla u|^{2p} \frac {\partial u}{\partial n}= f(u), x in \partial \Omega,t in (0,t^*)$, $u (x,0) = u_0( x)\geq 0, x in \Omega.$ p is some nonnegative parameter, n stands for the outward normal. It is shown that under certain conditions on the nonlinearities and data, blow-up will occur at some finite time. When blow-up does occur an lower bound for the blow-up time is obtained, by using a Sobolev-type inequality. An upper bound is also derived. References {BB} C.Bandle, H. Bruner, Blow-up in diffusion equations. A survey, J.Comput. Appl. Math. 97 (1998) 3--22. {PPV} L.E.Payne, G.A.Philippin, S. Vernier-Piro, Blow- up phenomena for a semilinear heat equation with nonlinear boundary condition I, Z. Angew. Math. Phys., 61 (2010) 999-1007. {PS1} L.E.Payne, P.W. Schaefer, Lower bound for blow-up time in parabolic problems under Neumann conditions, Applicable Analysis 85 (2006) 1301--1311. {QS} R Quittner and P. Souplet, Superlinear parabolic problems. Blow-up, global existence and steady states, Birkh\"auser Advanced Texts, Basel, (2007) {V} J. L. Vazquez, The problems of blow-up for nonlinear heat equations. Complete blow-up and avalanche formation, Rend. Mat. Acc. Lincei s. 9 15 (2004) 281--300. {W1} F.B.Weissler, Local existence and nonexistence for semilinear parabolic equations in $L^p$, Indiana Univ. Math. J. 29 (1980) 79--102 . {W2} F.B.Weissler, Existence and nonexistence of global solutions for a heat equation, Israel J.Math. 38 (1981) n.1-2, 29--40.
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