NUMERICAL QUENCHING FOR A NONLINEAR DIFFUSION EQUATION WITH SINGULAR BOUNDARY FLUX

In this paper, we study the semidiscrete approximation of the solution of a nonlinear diffusion equation with nonlinear source and singular boundary flux. We find some conditions under which the solution of the semidiscrete form quenches in a finite time and estimate its semidiscrete quenching time. We also establish the convergence of the semidiscrete quenching time to the theoretical one when the mesh size tends to zero. Finally, we give some numerical experiments for a best illustration of our analysis.

Gradient estimates for an orthotropic nonlinear diffusion equation

We consider a quasilinear degenerate parabolic equation driven by the orthotropic p-Laplacian. We prove that local weak solutions are locally Lipschitz continuous in the spatial variable, uniformly in time.

The process of magnetic flux penetration into superconductors

In the present paper the magnetic flux penetration dynamics of type-II superconductors in the flux creep regime is studied by analytically solving the nonlinear diffusion equation for the magnetic flux induction, assuming that an applied field parallel to the surface of the sample and using a power-law dependence of the differential resistivity on the magnetic field induction. An exact solution of nonlinear diffusion equation for the magnetic induction B(r, t) is obtained by using a well-known self-similar technique. We study the problem in the framework of a macroscopic approach, in which all length scales are larger than the flux-line spacing; thus, the superconductor is considered as a uniform medium.

NONLINEAR DIFFUSION EQUATION WITH A PERTURBED CONVECTIONTERM: POTENTIAL SYMMETRIES WITH RESPECT TO THE SECOND CONSERVATION LAW

We consider the nonlinear diffusion equation with a perturbed convection term. The potential symmetries for the exact equation with respect to the second conservation law are classified. It is found that these exist only in the linear case. It is further shown that no nontrivial approximate potential symmetries of order one exists for the perturbed equation with respect to the other conservation law.

A nonlinear diffusion equation with reaction localized in the half-line

<abstract><p>We study the behaviour of the solutions to the quasilinear heat equation with a reaction restricted to a half-line</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ u_t = (u^m)_{xx}+a(x) u^p, $\end{document} </tex-math></disp-formula></p> <p>$ m, p &gt; 0 $ and $ a(x) = 1 $ for $ x &gt; 0 $, $ a(x) = 0 $ for $ x &lt; 0 $. We first characterize the global existence exponent $ p_0 = 1 $ and the Fujita exponent $ p_c = m+2 $. Then we pass to study the grow-up rate in the case $ p\le1 $ and the blow-up rate for $ p &gt; 1 $. In particular we show that the grow-up rate is different as for global reaction if $ p &gt; m $ or $ p = 1\neq m $.</p></abstract>