Properties Of Solutions To Partial Differential Equations With Dynamical Boundary Conditions

In this paper,we study a class of initial boundary value problems for partial differential equations with dynamical boundary conditions.Different from the three kinds of classical boundary value problems,the boundary conditions considered in this paper include both spatial derivatives and time derivatives of unknown functions.More specifically,the boundary conditions we consider include critical Sobolev growth and pDirichlct-to-Neumann operator(p-DtN operator for short).Although there has been a lot of research on partial differential equations with dynamical boundary conditions,there are few studies on the initial boundary value elliptic problems with dynamical boundary conditions involving critical Sobolev exponent and Hardy potential.In this paper,the corresponding equations with three different indexes are studied.The results generalize and supplement the previous work.In addition,the extension method used in this paper to transform the nonlocal problem into a local problem with higher one-dimensional variation provides a research idea for parabolic equations with nonlocal operators.In this paper,the Laplace equation with dynamical boundary conditions involving critical Sobolev exponent and Hardy potential,the p-Laplace equation with dynamical boundary conditions involving critical Sobolev exponent and p-DtN operator,and the p-Laplace equation with dynamical boundary conditions involving critical exponent and Hardy potential are considered respectively.We use the energy method to prove the existence and decay estimates of the global solution and the finite time blow-up of the local solution.The relationship between the global solution and the steady-state solution is discussed.The regularity of the solution is improved by Moser iteration method.Finally,the concentration phenomenon of the solution is accurately described by using the concentration compactness principle.