Under the consideration that the non-local condition has a spatial impact to the study of the boundary values problems, we present a study of the existence and uniqueness of weak solution for nonlinear parabolic Bessel problem with Neumann integral conditions, in addition to part devoted to the proof of the finite time blow up solutions. Actually, in the case p ≥ 1, sufficient conditions of blow up of solutions can be established by Kaplan's method backed by the numerical results.

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In this paper, we examine a nonlinear hyperbolic equation with a nonlinear integral condition, where we prove the existence and the uniqueness of the linear problem by the Faedo-Galerkin method. By applying an iterative process to some significant results obtained for the linear problem, the existence and the uniqueness of the weak solution for the nonlinear problem are additionally examined.

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