Absence of global solutions to a class of nonlinear parabolic inequalities

M. Guedda¹

¹ LAMFA, CNRS UMR 6140 Faculté de Mathématiques et d'Informatique Université de Picardie Jules Verne 33, rue Saint-Leu 80039 Amiens, France

Colloquium Mathematicum, Tome 94 (2002) no. 2, pp. 195-220

Résumé

We study the absence of nonnegative global solutions to parabolic inequalities of the type $ u_t \geq -(-{\mit \Delta })^{{\beta /2}} u - V(x)u + h(x,t) u^{p}$, where $ (-{\mit \Delta })^{{\beta /2}}$, $0 \beta \leq 2 $, is the $ \beta /2 $ fractional power of the Laplacian. We give a sufficient condition which implies that the only global solution is trivial if $ p > 1 $ is small. Among other properties, we derive a necessary condition for the existence of local and global nonnegative solutions to the above problem for the function $ V $ satisfying $ V_+(x)\sim a | x| ^{-b}$, where $a \geq 0$, $b > 0$, $p > 1 $ and $ V_+(x):= \max\{V(x),0\}$. We show that the existence of solutions depends on the behavior at infinity of both initial data and $ h$. In addition to our main results, we also discuss the nonexistence of solutions for some degenerate parabolic inequalities like $ u_t \geq {\mit \Delta } u^m + u^p $ and $ u_t \geq {\mit \Delta }_p u + h(x,t)u^p$. The approach is based upon a duality argument combined with an appropriate choice of a test function. First we obtain an a priori estimate and then we use a scaling argument to prove our nonexistence results.

Détail

DOI : 10.4064/cm94-2-3

Keywords: study absence nonnegative global solutions parabolic inequalities type geq mit delta beta where mit delta beta beta leq beta fractional power laplacian sufficient condition which implies only global solution trivial small among other properties derive necessary condition existence local global nonnegative solutions above problem function satisfying sim b where geq max existence solutions depends behavior infinity initial addition main results discuss nonexistence solutions degenerate parabolic inequalities geq mit delta geq mit delta u approach based duality argument combined appropriate choice test function first obtain priori estimate scaling argument prove nonexistence results