Initial-boundary value problem for higher-orders nonlinear elliptic-parabolic equations with variable exponents of the nonlinearity in unbounded domains without conditions at infinity - M. M. Bokalo; O. V. Domanska (Математичні Студії)
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 $|x|→ +∞$ (for example, growth restriction of solution as $|x|→ +∞$, or the solution to belong to some functional spaces). Note, that we need some restrictions on the data-in behavior as $|x|→ +∞$ for the initial-boundary value problems for equations considered above to be solvable.