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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  

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... 

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... 

A moduli spase consists in classifying geometric objects up tp equivalence relations and its point are in 1-1 corrospondence with equivalence classes. One of the most important moduli spaces in algebraic geometry is «moduli space of stable n-pointed curves of genus zero» which has many applications in mathematics and theoretical physics.In [10] Knudsen shows that for every n 3 there is a smooth projective variety Xn that is a moduli space for stable n-pointed curves of genus zero.In this thesis we study moduli spaces and introduce Xn and its relation with cross ratios. This way we show Xn is smooth projective variety and a fine moduli space. [5] After that we study a paper by Sean Keel. [9]... 

Consider the following Two Reaction-Diffusion System with Predator-Prey interactions.
{ █(ut = Du Δu + f (u) "" bϕ(u)v x ∈ Ω ,t>0 ,@ @vt = Dv Δv + g(v) + cϕ(u)v x ∈ Ω ,t>0 ,)┤
The main purposes of Thesis are as follow: The effect of a protection zone in the diffusive Leslie predator–prey model, Non-existence of non-constant positive steady states of two Holling type-II predator–prey systems: Strong interaction case, In the first chapter, my work is devoted to investigate the change of behavior of diffusive Leslie predator–prey model with large intrinsic predator growth rate, when a simple protection zone Ω_0 for the prey is introduced. In other word, the...