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This thesis is based on articles [18,15]. Zhao [18] has studied a free boundary problem modeling the growth of tumors with drug application. In this model live cells are two kindes: proliferative cells and quiescent cells. This model consists of two nonlinear second-order parabolic equations describing the diffusion of nutrient and drug concentration, and three nonlinear first-order hyperbolic equations describing the evolution of proliferative cells, quiescent cells and dead cells. He has proved that this free boundary problem has a unique global solution. Tao and Chen [15] have studied another free boundary problem modelling the growth of an avascular tumour with drug application. The... 

In this thesis, two kind of Kirchhoff type equations are investigated. for each of them we prove existence and uniqueness of weak solution by letting hypothesizes on problems functions and initial conditions and using Faedo-Galerkin method and some theorems in functional analysis. It is should be mentioned that one of the problems has unique strong solution according to compactness theorems. In the nest step stability of these equations are investigated. for this purpose it is showed that total energy of systems will tend to zero as time goes to infinity. Furthermore at the end of each chapter numerical results are presented to illustrate accuracy of obtained theoretical results. Finally we... 

We consider the following viscous Hamilton-Jacobi Equations for :

The aim of this paper is to investigate relations between:
(i) The existence of global solutions,
(ii) The existence of stationary solutions (with gradient possibly singular on the boundary), and we obtain precise description of these relations. Namely, (i) imply that (ii), and this case all global solutions converge uniformly to unique stationary solutions. In the redial case, we prove converse of this result. Moreover, for certain smooth function we obtain the existence of global classical solutions with gradient blow-up in infinite time. For or for Cauchy problem, we establish similar relations. Our... 

This thesis is concerned with the Boussinesq-Burgers system which models the propagation of bores by combing the dissipation, Dispasion and nonlinearity . we establish the global existence and asymptotical behavior of classical solutions of the initial value and Dirichlet boundary problem of the Boussinesq-Burgers system with the help of a lyapunov functional and the technique of Moser itrtation. Particularly we show that the solution converges to the constant stationary solution as time tends to infinity. Furthermore if we have real domain or non-hemogenous boundary condition then we will arrive constant stationary solution as time tends to infinity  

In this thesis we investigating two models of cancer invasion .First, a general mathematical model of cancer invasion is presented. In this model there are three factors: tumor cell, extracellular matrix and enzyme. The model consists of a parabolic partial differential equation (PDE) describing the evolution of tumor cell density , an ordinary differential equation modeling of extracellular matrix and a parabolic PDE governing the evolution of the matrix degrading enzyme concentration. This model is investigated in two special versions for existence and uniqueness of global solutions. In the first model we neglect the remodeling term, this model is named the chemotaxis-haptotaxis model.... 

This thesis studies a haptotaxis system proposed as a model for oncolytic virotherapy , accounting for interaction between uninfected cancer cells , infected cancer cells , extracellular matrix (ECM) and oncolytic virus . In addition to random movement , both uninfected and infected tumor cells migrate haptotactically toward higher ECM densities; moreover , besides degrading the non-diffusible ECM upon contact the two cancer cell populations are subject to an infection-induced transition mechanism driven by virus particles which are released by infected cancer cells ,and which assault the uninfected part of the tumor .The main results assert global classical solvability in an associated...