Absence of global solutions to a class of nonlinear parabolic inequalities

Volume 94 / 2002

M. Guedda Colloquium Mathematicum 94 (2002), 195-220 MSC: 35K55, 35K65. DOI: 10.4064/cm94-2-3


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.


  • M. GueddaLAMFA, CNRS UMR 6140
    Faculté de Mathématiques et d'Informatique
    Université de Picardie Jules Verne
    33, rue Saint-Leu
    80039 Amiens, France

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