The rate of spatial decay of nonnegative solutions of nonlinear parabolic equations and inequalities

Abstract:

Let $L$ be a uniformly parabolic linear partial differential operator. We show that nonnegative solutions of the differential inequality $Lu \leq c(u + |\nabla u|)$ on ${{\mathbf {R}}^n} \times (0,T)$ for which $u(x,T) = {\mathbf {0}}(\exp {\text {(}} - \delta |x{|^2}))$ must be identically zero if the constant $\delta$ is sufficiently large. An analogous result is given for nonlinear systems.