Initial-boundary value problem for higher-orders nonlinear elliptic-parabolic equations with variable exponents of the nonlinearity in unbounded domains without conditions at infinity

Abstract

Initial-boundary value problems for parabolic and elliptic-parabolic (that is degenerated parabolic) equations in unbounded domains with respect to the spatial variables were studied by many authors. It is well known that in order to guarantee the uniqueness of the solution of the initial-boundary value problems for linear and some nonlinear parabolic and elliptic-parabolic equations in unbounded domains we need some restrictions on behavior of solution as $|x|\to +\infty$ (for example, growth restriction of solution as $|x|\to +\infty$, or the solution to belong to some functional spaces).Note, that we need some restrictions on the data-in behavior as$|x|\to +\infty$ for the initial-boundary value problemsfor equations considered above to be solvable.
 However, there are nonlinear parabolic equations for whichthe corresponding initial-boundary value problems are uniquely solvable withoutany conditions at infinity.
 We prove the unique solvability of the initial-boundary value problemwithout conditions at infinity for some of the higher-orders anisotropic parabolic equationswith variable exponents of the nonlinearity. A priori estimate of the weak solutionsof this problem was also obtained. As far as we know, the initial-boundary value problem for the higher-orders anisotropic elliptic-parabolic equations with variable exponents of nonlinearity in unbounded domains were not considered before.