On Parabolic Equations with Mixed Dirichlet–Robin Type Boundary Conditions in a Non-rectangular Domain

Abstract

In this paper, we prove well-posedness and smoothness results in anisotropic Sobolev spaces for the solutions of boundary value problems with Dirichlet-Robin type boundary conditions for second-order parabolic equations in non-rectangular domains.Mathematics Subject Classification. 35K05, 35K20.

Blow-up of solutions for nonlinear parabolic equation with nonlocal source and nonlocal boundary condition

“…They established local existence, global existence and nonexistence of solutions and discussed the blow up properties of solutions. Similar results can be seen in [23,24,[26][27][28][29][30].…”

“…On the other hand, due to the appearance of the nonlocal boundary condition, the properties of solution heavily depend on the weight function uðx; yÞ as well. Motivated by the above cited works and the references [18,22,[25][26][27][28][29][30], in this paper we deal with a nonlinear parabolic equation subject to nonlocal boundary conditions and with nonlocal sources. By means of a sub-super-solution method, we obtained conditions which guarantee the solutions having global existence or blowing up in finite time (see Section 3 for details).…”

Parabolic equations with Cauchy–Dirichlet boundary conditions in a non-regular domain of ℝN+1

“…In Sadallah [18], the same problem has been studied for a 2 -parabolic operator in the case of one space variable. Further references on the analysis of parabolic problems in non-cylindrical domains are: Savaré [19], Aref'ev and Bagirov [3], Ho mann and Lewis [8], Labbas, Medeghri and Sadallah [13,14], Alkhutov [1,2] and Khelou et al [9][10][11][12]. There are many other works concerning boundary-value problems in non-smooth domains (see, for example, Grisvard [7] and the references therein).…”

Existence of solution to parabolic equations with mixed boundary condition on non-cylindrical domains

“…[11]), parabolic equation with Robin type boundary condition (cf. [12]) in non-cylindrical domains and behavior of solutions to the initial-boundary value problems of nonlinear equations (cf. [16] and [29]) are studied.…”