semilinear elliptic and parabolic inequalities
Elliptic and parabolic differential inequalities
The author establishes comparison results for some semilinear elliptic and parabolic inequalities of the forms NEWLINE\[NEWLINE-\Delta \mu+\omega\mu\leq f(x,u)\text{ a.a. }x\in\Omega,\;u\in H^1_0(\Omega),\quad \Delta u\in L^2(\Omega),NEWLINE\]NEWLINE respectively NEWLINE\[NEWLINE\begin{aligned} & \frac{\partial \mu}{\partial t}-\Delta u\leq f(x,t,u)\text{ a.a. }x\in \Omega,\text{ all } t>0,\quad u(x,0)\leq \mu_0(x)\text{ a.a. }x\in\Omega,\\& u(\cdot,t)\in H^1_0(\Omega),\;\Delta u(\cdot,t)\in L^2(\Omega)\text{ for all }t>0,\;\frac{\partial \mu}{\partial t}\in L^1(0,T;L^2(\Omega)).\end{aligned}NEWLINE\]NEWLINE Let us mention that similar results for elliptic equations are presented by \textit{D. Gilbarg} and \textit{N. S. Trudinger} [Elliptic partial differential equations of second order, Springer, (1983; Zbl 0562.35001], but they need some continuity and differentiability for the function \(f\).
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