The nonpositivity of solutions to pseudoparabolic equations

Abstract:

Conditions are given on the nonnegative data $\phi (x)$ and $g(x,t)$ such that solutions of the pseudoparabolic inequality $P[u] = (L - I){u_t} + Lu \leqslant 0$ in $Dx(0,\tau )$ \[ \begin {array}{*{20}{c}} {u(x,0) = \phi (x),} \hfill & {x \in D,} \hfill \\ {u(x,t) = g(x,t),} \hfill & {x \in D \times (0,\tau ),} \hfill \\ \end {array} \] satisfy $u(x,t) \geqslant 0$ in $D \times (0,\tau )$. Here D is an open set in ${{\mathbf {R}}^n}$ and L is a second order elliptic differential operator. A counterexample is provided to show that this condition is in a sense necessary. The result implies that solutions $P[u] = 0$ do not in general satisfy a maximum principle.