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This thesis is devoted to prove the existance of solution and structure of solu­ tion for some partial differential equations by using some modern topological and variational thechniques. Taking direction from the literature, this thesis is interested in existence, uniqueness, blow-up in finite time for some evolution equations, multiplicity and radial solution for certain elliptic partial differential equations.-Employing Fibrering, Galerkins, Mountain Pass-Lemma and lions com­ pactness Lemma are sharp in this thesis to prove the exsistence and multiplicity of solutions and overcome lack of compactness in some cases  

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 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.... 

In this thesis, three mathematical models are considered for the viral dynamics of HIV-1. The first model is an HIV infection model with saturation infection and intracellular delay, which forms a three-dimensional differential equations system, the second model includes an eclipse stage of infected cells, The viral dynamics of this model is described by four nonlinear ordinary differential equations, and finally, we study a delayed six-dimensional HIV model with CTL immune response, in fact, the main issue is the analysis of the second model (model including an eclipse stage for the infected cells).In this thesis is obtained sufficient conditions for persistence or eradication of the... 

We Consider the Chemotaxi Systems in a smooth bounded domain as follow:{ █(ut = Δu-χ∇.(u∇v)+ f (u) x ϵ Ω ,t>0 @ @τ vt =Δv-v+u x ϵ Ω ,t>0)┤ Where χ∈ and f(u) =Au - Buα generalizes the logistic function with A≥0, B>0 and α>1. First for τ=0, global existence of such solutions for any nonnegative initial data is proved under the assumption that . Moreover, boundedness properties of the constructed solutions are studied. Next we assume that 2=α, τ>0 and we consider nonnegative solutions of the Neumann
Boundary value problem for the chemotaxis system above in a smooth bounded convex domain . We will see that if B is sufficiently large then for all sufficiently smooth initial data the... 

In this thesis, a complete analysis of a general within-host models of malaria is done. This model generalizes the models in epidemiological literature. In this study, we propose another equation for immune effectors reaction. In this thesis, we find out that the global stability of disease free equilibrium is obtained when the reproduction number R0 < 1. When R0 ≥ 1, at least one endemic equilibrium exists. The local and global asymptotic stability is investigated. Finally, numerical simulations are done to illustrate the influence of immune effectors reaction  

In optimal control,local quadratic growth estimation in case of continuous control functions, and for bang–bang optimal controls when the state system is linear, has been obtained. The paper provides a generalization of the latter result to bang–bang optimal control problems for systems which are affine-linear w.r.t. the control but depend nonlinearly on the state.Local quadratic growth in terms of L1 norms of the control variation are obtained under appropriate structural and second-order sufficient optimality conditions  

In season 1, we introduce a mathematical model of the in vivo progression of Alzheimer’s disease with focus on the role of prions in memory impairment. Our model consists of differential equations that describe the dynamic formation of Aβ -amyloid plaques based on the concentrations of Aβ oligomers, PrPC proteins, and the Aβ-×-PrPC complex, which are hypothesized to be responsible for synaptic tox- icity. We prove the well posedness of the model and provided stability results for its unique equilibrium, when the polymerization rate of β-amyloid is constant and also when it is described by a power law. In seson 2, We consider the existence and uniqueness of solutions of an initial-boundary... 

Dengue is one of the most important infectious diseases in the world. This disease is a viral infection that is transmitted to humans through the bite of a mosquito called Aedes aegypti. For this reason, geographical regions infected with this type of mosquito are at risk of Dengue outbreak. In this thesis, we first present a mathematical model describing the geographical spread of Dengue disease, which includes the movement of both the human population and the winged mosquito population. This model is derived from a mixed system of partial and ordinary differential equations. Our proposed model has the ability to consider the possibility of asymptomatic infection, so that the presence of... 

In this thesis, a new non-linear control synthsis technique (θ - D) approximation) is discussed. This approach achieves suboptimal solutions to a class of non-linear optimal control problems characterized by a quadratic cost function and a plant model that is a_ne in control. An approximate solution to the Hamilton-Jacobi-Bellman (HJB) equation is sought by adding perturbations to the cost function. By manipulating the perturbation terms both asymptotic stability and suboptimality properties are obtained. The new technique overcomes the large-control-for-large-initial-states problem that occurs in some other Taylor series expansion based methods. Also this method does not require excessive... 

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... 

In this thesis we study the problem of existence and multiplicity of positive periodic solution to the scalar ODE , , where is a positive function on , super linear at zero and sub linear at infinity, and is a -periodic and sign indefinite weight with negative mean value. We first show the nonexistence of solution for some classes of nonlinearities when is small. Then, using critical point theory, we prove the existence of at least two positive -periodic solutions for large. Then, we prove the existence of a pair of positive -periodic solutions as well as the existence of positive sub harmonic solutions of any order for the scalar second order ODE where is same as above,... 

In this thesis , we consider the generalized lienard system (dx/dt=1/(a(x)) [h(y)-F(x)])¦(dy/dt=-a(x)g(x) )and under suitable assumptions on a , F , g , h we obtain sufficient and necessary conditions for the intersection of all orbits with the vertical isocline y=F(x) . using these conditions we give some sufficient condition for the oscillation of solutions . then existence and uniqueness of periodic solutions for a kind of lienard equation with a deviating argument are studied . finally we study existence and uniqueness of limit cycles for the generalized lienard system(x ̇=ϕ(y)-F(x))¦(y ̇=-g(x))
 

Moving of living organisms appears in many interesting problems, e.g. the growth of bacteria colonies, tumor growth, wound healing, color patterns of animals and etc. There are many ways to model such problems and PDE theory is widely used to investigate these problems. In this thesis, we study two well-known classic models. First, macroscopic “Keller–Segel” model and then kinetic “Othmer–Dunbar–Alt” System. Since these models have a nice behavior in two dimensions that they don’t have in other dimensions, we propose a way to alter them such that they behave in this way in all dimensions. Also none of the known models have the suitable dynamics in one dimension, so our model has the property... 

The predator – prey model is arises when the growing rate of one type decreased and the growing rate of another type increased. In this project, in chapters 2,3,4 and 5 we will describe the existence of periodic solutions for predator-prey systems and introduce some classical techniques for finding this solutions which the most important of them is the continuation theorem, for using this theorem we must find the open set Ω which by the conditions on Fredholm mapping L and the mappings Q , N we solve the equation Lx=Nx . In chapter 2 we study the periodic solutions of the below systemx2(k)






 

The aim of this work is to study the optimal exponent p to have solvability of semilinear biharmonic problem with a singular term in a smooth and bounded domain such that contains origin in Euclidean space with dimension greater than 4. The singular term is related to the Hardy inequality. First of all, it is not difficult to show that any positive supersolution of problem is unbounded near the origin and then additional hypotheses on p are needed to ensure existence of solutions. We will say that problem blows up completely if the solutions to the truncated problems (with a bounded weight instead of the Hardy singularity) tend to infinity for every x in domain as n goes infinity. The main... 

IN this upper ,we consider a reaction –diffusion predator-prey model with stage –structure, holling type –ll functional response. Nonlocal spatial impacat and harvesting The stability of the equilibria is investigated. Furthermore, by the cross-iteration schauder fixed point theorem, we deduce the existence of traveling wave solution which connects the zero solution and the positive constant eguilibrium  

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...