Existence of unbounded solutions to parabolic equations with functional dependence

Abstract

The Cauchy problem for nonlinear parabolic differential-functional equations is considered. Under natural generalized Lipschitz-type conditions with weights, the existence and uniqueness of unbounded solutions is obtained in three main cases: (i) the functional dependence u(·); (ii) the functional dependence u(·) and ∂xu(·); (iii) the functional dependence u(·) and the pointwise dependence ∂xu (t, x).

Existence of solutions with exponential growth for nonlinear differential-functional parabolic equations

“…Existence results for parabolic equations with functionals of the unknown function and its spatial first-order derivatives were considered in [14] as fixed points of suitable integral operators. Our research into existence and uniqueness of such solutions for various parabolic differential-functional problems has gradually developed by the use of iterative methods or the Banach contraction principle (see [3,5,10,11]). In all these works estimates of the fundamental solutions based on [6,9,12,18] are applied.…”