We consider the inverse problem of identification degenerate diffusion coefficient form $x^\alpha a(x)$ in a one dimensional parabolic equation by some extra data. first prove energy methods uniqueness and Lipschitz stability results for constant $a$ power $\alpha$ knowing an interior data at time. On other hand, we obtain result general coefficients $a(x)$ also boundary on side space interval....

<p style='text-indent:20px;'>In this paper, we consider the following degenerate/singular parabolic equation</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{align*} u_t -(x^\alpha u_{x})_x - \frac{\mu}{x^{2-\alpha}} u = 0, \qquad x\in (0,1), \ t \in (0,T), \end{align*} $\end{document} </tex-math></...

In this paper, we prove the existence of asymptotic speed solutions to fully nonlinear, possibly degenerate parabolic partial differential equations in a general setting. We then give some explicit examples setting and study further properties for each equation. Some numerical results concerning are presented.

We study a degenerate nonlinear parabolic equation with moving boundaries which arises in the study of the technique of contour enhancement in image processing. In order to obtain mass concentration at the contour, singular data must be imposed at the free boundary, leading to a non-standard free boundary problem. Our main results are: (i) the well-posedness for the singular problem, without mo...

Using the theory of uniform global attractors of multivalued semiprocesses, we prove the existence of a uniform global attractor for a nonautonomous semilinear degenerate parabolic equation in which the conditions imposed on the nonlinearity provide the global existence of a weak solution, but not uniqueness. The Kneser property of solutions is also studied, and as a result we obtain the connec...

This paper concerns a superlinear parabolic equation, degenerate in the time derivative. It is shown that the solution may blow up in finite time. Moreover it is proved that for a large class of initial data blow-up occurs at the boundary of the domain when the nonlinearity is no worse than quadratic. Various estimates are obtained which determine the asymptotic behaviour near the blow-up. The ...

We prove local integral (entropy) estimates for nonnegative solutions of the fourth order degenerate parabolic equation u t + div(u n ru) = 0 in space dimensions two and three. These estimates enable us to show that solutions have nite speed of propagation if n 2 (1 8 ; 2) and that the support cannot shrink if the growth exponent n is larger than 3=2. In addition, we prove decay estimates for s...

We consider a nonlinear 4th-order degenerate parabolic partial differential equation that arises in modelling the dynamics of an incompressible thin liquid film on the outer surface of a rotating horizontal cylinder in the presence of gravity. The parameters involved determine a rich variety of qualitatively different flows. We obtain sufficient conditions for finite speed of support propagatio...

We study the gradient flow of the Riemannian functional F(g) := ∫ M |Rm| . This flow corresponds to a fourth-order degenerate parabolic equation for a Riemannian metric. We prove that, as with the Ricci flow, the degeneracies may be accounted for entirely by diffeomorphism flow, and hence we show short-time existence using the DeTurck method. We prove L2 derivative estimates of Bernstein-Bando-...

In this paper we study controllability properties for semilinear degenerate parabolic equations with nonlinearities involving the first derivative in a bounded domain of R. Due to degeneracy, classical null controllability results do not hold in general. Thus we investigate results of ’regional null controllability’, showing that we can drive the solution to rest at time T on a subset of the sp...