Solvability issues for a class of convolution type nonlinear integral equations in $\mathbb {R}^n$

We study a class of nonlinear multidimensional integral equations of convolution type. This class of equations is directly applied in the p-adic theory of open-closed strings. We prove the existence of an n-parametric family of nontrivial continuous bounded solutions and establish certain properties of the constructed solutions: monotonicity in each argument, limit relations, and integral asymptotics. The solutions are used to study a nonlinear problem for the multidimensional heat equation. At the end of the paper we give example of such equations, which are of independent theoretical and practical interest.