无界区域上无无穷条件非线性变指数高阶非线性抛物型方程的初边值问题

M. Bokalo
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引用次数: 1

摘要

许多作者研究了无界区域上关于空间变量的抛物型方程的初边值问题。众所周知,为了保证无界域上线性和某些非线性抛物型方程的初边值问题解的唯一性,需要对解的行为作$|x|\to +\infty$的限制(例如,解的生长限制为$|x|\to +\infty$,或解属于某些泛函空间)。注意,对于上述抛物型方程初边值问题的可解性,我们需要对数据入行为$|x|\to +\infty$进行一些限制。然而,存在非线性抛物型方程,其相应的初边值问题在无穷远处无任何条件下是唯一可解的。变指数非线性微分方程作为数学模型出现在各种物理过程中。特别地,这些方程描述了在温度场变化的情况下,电流在导体中的流动、图像恢复过程和电流。许多著作对变指数非线性微分方程进行了深入的研究。在这些研究中使用了相应的Lebesgue和Sobolev空间的推广。本文证明了一类非线性变指数高阶各向异性抛物型方程无穷远处无条件初边值问题的唯一可解性。给出了该问题的广义解的先验估计。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
INITIAL-BOUNDARY VALUE PROBLEM FOR HIGHER-ORDERS NONLINEAR PARABOLIC EQUATIONS WITH VARIABLE EXPONENTS OF THE NONLINEARITY IN UNBOUNDED DOMAINS WITHOUT CONDITIONS AT INFINITY
Initial-boundary value problems for parabolic equations in unbounded domains with respect to the spatial variables were studied by many authors. As is well known, to guarantee the uniqueness of the solution of the initial-boundary value problems for linear and some nonlinear parabolic equations in unbounded domains we need some restrictions on solution's behavior as $|x|\to +\infty$ (for example, solution's growth restriction as $|x|\to +\infty$, or belonging of solution to some functional spaces). Note that we need some restrictions on the data-in behavior as $|x|\to +\infty$ to solvability of the initial-boundary value problems for parabolic equations considered above. However, there are nonlinear parabolic equations for which the corresponding initial-boundary value problems are unique solvable without any conditions at infinity. Nonlinear differential equations with variable exponents of the nonlinearity appear as mathematical models in various physical processes. In particular, these equations describe electroreological substance flows, image recovering processes, electric current in the conductor with changing temperature field. Nonlinear differential equations with variable exponents of the nonlinearity were intensively studied in many works. The corresponding generalizations of Lebesgue and Sobolev spaces were used in these investigations. In this paper we prove the unique solvability of the initial--boundary value problem without conditions at infinity for some of the higher-orders anisotropic parabolic equations with variable exponents of the nonlinearity. An a priori estimate of the generalized solutions of this problem was also obtained.
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