Liouville theorems for nonlinear parabolic equations of second order
The authors consider entire solutions, \(u= u(x, t)\), (i.e. solutions defined for all \((x, t)\in \mathbb{R}^n\times \mathbb{R}\)) to the nonlinear, parabolic equation \[ \sum^n_{i, j= 1} a_{ij}(x, t) {\partial^2\over \partial x_i \partial x_j} u(x, t)+ H(x, t, u(x, t), \partial u(x, t)/\partial x_1,\dots, \partial u(x, t)/\partial x_n)- u_t(x, t)= 0. \] The matrix \(a(x, t)= (a_{ij}(x, t))\) is symmetric and positive definite, it approaches the identity matrix at infinity, and satisfies the Hölder condition \[ |z(x, t)- a(y, s)|\leq K\Biggl[{|x- y|^2+ |t- s|\over 1+ |x|^2+ |y|^2+ |t|+ |s|}\Biggr]^\alpha, \] where \(K\geq 0\), \(0< \alpha\leq 1\) are constants. The function \(H= H(x, t, u, z)\) is defined on \(\mathbb{R}^n\times \mathbb{R}\times \mathbb{R}\times \mathbb{R}^n\), is measurable in \((x, t)\) and Lipschitz continuous in \((u, z)\). The first part of the paper is devoted to obtaining a priori bounds on solutions to the inhomogeneous heat equation and then the general linear equation. These bounds are coupled with the Schauder continuation method to study solutions to the nonlinear equation. The authors prove that, under certain hypotheses, given a polynomial solution \(P(x, t)\) of the heat equation with a certain allowable growth rate, there exists a unique solution \(u(x, t)\) to \((*)\) such that \(u- P\) vanishes at infinity. When specialized to the linear, homogeneous parabolic equation with no zero-order term, this result yields the Liouville type theorem that a bounded, entire solution be constant. The proofs of these results are given in detail and are computationally intense.
scientific article; zbMATH DE number 840306