A Liouville comparison principle for solutions of semilinear parabolic second-order partial differential inequalities

Abstract

We obtain a new Liouville comparison principle for entire weak solutions $(u,v)$ of semilinear parabolic second-order partial differential inequalities of the form $$u_t-\mathcal{L}u-|u{|}^{q-1}u\ge v_t-\mathcal{L}v-|v{|}^{q-1}v(\ast )$$ in the half-space $\mathbb{S}={\mathbb{R}}_{+}^1\times {\mathbb{R}}^n$. Here $n\ge 1$, $q>0$ and $$\mathcal{L}=\sum _{i,j=1}^n\frac{\partial }{\partial x_i}[a_{ij}(t,x)\frac{\partial }{\partial x_j}],$$ where $a_{ij}(t,x)$, $i,j=1,...,n$, are functions defined, measurable and locally bounded in $\mathbb{S}$, and such that $a_{ij}(t,x)=a_{ji}(t,x)$ and $$\sum _{i,j=1}^n{a}_{ij}(t,x){\xi }_i{\xi }_j\ge 0$$ for almost all $(t,x)\in \mathbb{S}$ and all $\xi \in {\mathbb{R}}^n$. The critical exponents in the Liouville comparison principle obtained, which responsible for the non-existence of non-trivial (i.e., such that $u\not\equiv v$) entire weak solutions to (*) in $\mathbb{S}$, depend on the behaviour of the coefficients of the operator $\mathcal{L}$ at infinity. As direct corollaries we obtain a new Fujita comparison principle for entire weak solutions $(u,v)$ of the Cauchy problem for the inequality (*), as well as new Liouville-type and Fujita-type theorems for non-negative entire weak solutions $u$ of the inequality (*) in the case when $v\equiv 0$. All the results obtained are new and sharp.