General Decay Of Solutions For A Porous-elastic System With Memory

Partial Differential Equation,as a independent branch of mathematics,has a history of more than 200 years from the physical and geometric problem to today,including the regularity well-posedness,stability,controllability,decay and blow-up of elliptic,hyperbolic,parabolic equations.In the paper,based on the properties of the solution,general decay of the porous elastic system with memory is studied by the multiplier method.It is mentioned that porous elastic system is the hyperbolic equation.In the first chapter,the introduction explains the origin of the theory of porous elasticity and the properties of solutions for porous elastic system with different damping.Next,there are the following important formula in the preliminaries.In the second chapter,we are discussed with the memory type porous-elastic system with Neumann-Dirichlet boundary condition and equal or unequal of the wave speeds.We establish a general decay result,for which exponential and polynomial decay results are special cases.And we prove the result depending on the kernel of the memory term and the wave speeds of the system.In the third chapter,we are concerned with two memory-type porous-elastic system with Dirichlet-Dirichlet boundary condition.It is well-known that the unique dissipation given by the memory term is strong enough to stabilize the system,depending on the kernel of the memory term and the wave speeds of the system.In contrary,we prove that a general decay result of solutions depending only on the kernel of the memory term.