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In this paper, semi-linear parabolic equation with integral boundary condition of second type is investigated. The existence, uniqueness and Blow-up of weak solutions in finite time are established. The proof is proceeds in two steps; using the variable separation method for the solvability of the linear cas and applying an iterative process and a priori estimate, we prove the existence, uniqueness of the weak solution of the semilinear problem. Finally, we study a blow-up of solution in finite time for a super-linear problem by using eigen functions method introduced by Kaplan.

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This work is devoted to studying a viscoelastic Kirchhoff-type equation with nonlinear boundary damping-source interactions in a bounded domain. Under suitable assumptions on the kernel function g, density function, and M, the global existence and general decay rate of solution are established. Moreover, we prove the finite time blow-up result of solution with negative initial energy.