We study a boundary value problem with multivariables integral type condition for a class of parabolic equations. We prove the existence, uniqueness, and continuous dependence of the solution upon the data in the functional wieghted Sobolev spaces. Results are obtained by using a functional analysis method based on two-sided a priori estimates and on the density of the range of the linear operator generated by the considered problem.

Abstract. The main objective of this paper is to establish some new explicit bounds for nonlinear integral inequalities of Wendroff type with continuous and weakly singular kernel, which generalized some known inequalities for functions in two variables and can be furnished a handy tool for the study of qualitative as well as quantitative properties of solutions of nonlinear differential equations. Some applications are also given to illustrate the usefulness of our results.Mathematics subject classification (2010): 42B20, 26D07, 26D15.

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In this paper we are concerned with an integrodifferential problem which arises for instance when we study a Cauchy-type fractional differential equation. This problem involves a convolution of a kernel with a nonlinear function of the solution together with its derivatives up to order two. For ordinary (third order) differential equations the kernel is regular while in our case it is singular and nonintegrable. Combining a desingularization technique due to the second author with some other estimations, we find bounds for solutions of the problem with different nonlinearities. Our results are illustrated by an application to fractional differential equations.