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Register a new userOn a nonlocal degenerate parabolic problem
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Conditions for the existence and uniqueness of weak solutions for a class of nonlinear nonlocal degenerate parabolic equations are established. The asymptotic behaviour of the solutions as time tends to infinity are also studied. In particular, the finite time extinction and polynomial decay properties are proved.
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The aim of this paper is the numerical study of a class of nonlinear nonlocal degenerate parabolic equations. The convergence and error bounds of the solutions are proved for a linearized Crank-Nicolson-Galerkin finite element method with polynomial approximations of degree $kgeq 1$. Some explicit solutions are obtained and used to test the implementation of the method in Matlab environment.
We study a Neumann type initial-boundary value problem for strongly degenerate parabolic-hyperbolic equations under the nonlinearity-diffusivity condition. We suggest a notion of entropy solution for this problem and prove its uniqueness. The existence of entropy solutions is also discussed.
In this paper, we consider the asymptotic behavior of the nonlocal parabolic problem [ u_{t}=Delta u+displaystylefrac{lambda f(u)}{big(int_{Omega}f(u)dxbig)^{p}}, xin Omega, t>0, ] with homogeneous Dirichlet boundary condition, where $lambda>0, p>0$, $f$ is nonincreasing. It is found that: (a) For $0<pleq1$, $u(x,t)$ is globally bounded and the unique stationary solution is globally asymptotically stable for any $lambda>0$; (b) For $1<p<2$, $u(x,t)$ is globally bounded for any $lambda>0$; (c) For $p=2$, if $0<lambda<2|partialOmega|^2$, then $u(x,t)$ is globally bounded, if $lambda=2|partialOmega|^2$, there is no stationary solution and $u(x,t)$ is a global solution and $u(x,t)toinfty$ as $ttoinfty$ for all $xinOmega$, if $lambda>2|partialOmega|^2$, there is no stationary solution and $u(x,t)$ blows up in finite time for all $xinOmega$; (d) For $p>2$, there exists a $lambda^*>0$ such that for $lambda>lambda^*$, or for $0<lambdaleqlambda^*$ and $u_0(x)$ sufficiently large, $u(x,t)$ blows up in finite time. Moreover, some formal asymptotic estimates for the behavior of $u(x,t)$ as it blows up are obtained for $pgeq2$.
In this paper we study a nonlocal diffusion problem on a manifold. These kind of equations can model diffusions when there are long range effects and have been widely studied in Euclidean space. We first prove existence and uniqueness of solutions and a comparison principle. Then, for a convenient rescaling we prove that the operator under consideration converges to a multiple of the usual Heat-Beltrami operator on the manifold. Next, we look at the long time behavior on compact manifolds by studying the spectral properties of the operator. Finally, for the model case of hyperbolic space we study the long time asymptotics and find a different and interesting behavior.
We consider the Dirichlet problem for a class of elliptic and parabolic equations in the upper-half space $mathbb{R}^d_+$, where the coefficients are the product of $x_d^alpha, alpha in (-infty, 1),$ and a bounded uniformly elliptic matrix of coefficients. Thus, the coefficients are singular or degenerate near the boundary ${x_d =0}$ and they may not locally integrable. The novelty of the work is that we find proper weights under which the existence, uniqueness, and regularity of solutions in Sobolev spaces are established. These results appear to be the first of their kind and are new even if the coefficients are constant. They are also readily extended to systems of equations.
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