Optimal decay rate for degenerate parabolic equations on noncompact manifolds

We consider an initial value problem for a doubly degenerate parabolic equation in a noncompact Riemannian manifold. The geometrical features of the manifold are coded in either a Faber-Krahn inequality or a relative Faber-Krahn inequality. We prove optimal decay and space-time local estimates of solutions. We employ a simplified version of the by now classical local approach by DeGiorgi, Ladyzhenskaya-Uraltseva, DiBenedetto which is of independent interest even in the euclidean case

Solving Nonlinear Parabolic Equations by a Strongly Implicit Finite-Difference Scheme

We discuss the numerical solution of nonlinear parabolic partial differential equations, exhibiting finite speed of propagation, via a strongly implicit finite-difference scheme with formal truncation error $\mathcal{O}\left[(\Delta x)^2 + (\Delta t)^2 \right]$. Our application of interest is the spreading of viscous gravity currents in the study of which these type of differential equations arise. Viscous gravity currents are low Reynolds number (viscous forces dominate inertial forces) flow phenomena in which a dense, viscous fluid displaces a lighter (usually immiscible) fluid. The fluids may be confined by the sidewalls of a channel or propagate in an unconfined two-dimensional (or axisymmetric three-dimensional) geometry. Under the lubrication approximation, the mathematical description of the spreading of these fluids reduces to solving the so-called thin-film equation for the current's shape $h(x,t)$. To solve such nonlinear parabolic equations we propose a finite-difference scheme based on the Crank--Nicolson idea. We implement the scheme for problems involving a single spatial coordinate (i.e., two-dimensional, axisymmetric or spherically-symmetric three-dimensional currents) on an equispaced but staggered grid. We benchmark the scheme against analytical solutions and highlight its strong numerical stability by specifically considering the spreading of non-Newtonian power-law fluids in a variable-width confined channel-like geometry (a "Hele-Shaw cell") subject to a given mass conservation/balance constraint. We show that this constraint can be implemented by re-expressing it as nonlinear flux boundary conditions on the domain's endpoints. Then, we show numerically that the scheme achieves its full second-order accuracy in space and time. We also highlight through numerical simulations how the proposed scheme accurately respects the mass conservation/balance constraint.Comment: 36 pages, 9 figures, Springer book class; v2 includes improvements and corrections; to appear as a contribution in "Applied Wave Mathematics II

Finite speed of propagation for the thin-film equation and other higher-order parabolic equations with general nonlinearity

Abstract. We prove the property of finite speed of propagation for degenerate parabolic equations of order 2m ≥ 2, when the nonlinearity is of general type, and not necessarily a power function. We also give estimates of the growth in time of the interface bounding the support of the solution. In the case of the thin film equation, with non power nonlinearity, we obtain sharp results, in the range of nonlinearities we consider. Our optimality result seems to be new even in the case of power nonlinearities with general initial data. In the case of the Cauchy problem for degenerate equations with general m, our main assumption is a suitable integrability Dini condition to be satisfied by the nonlinearity itself. Our results generalise Bernis’ estimates for higher order equations with power structures. In the case of second order equations we also prove L ∞ estimates of solutions

Asymptotic Estimates for the p-Laplacian on Infinite Graphs with Decaying Initial Data

We consider the Cauchy problem for the evolutive discrete p-Laplacian in infinite graphs, with initial data decaying at infinity. We prove optimal sup and gradient bounds for nonnegative solutions, when the initial data has finite mass, and also sharp evaluation for the confinement of mass, i.e., the effective speed of propagation. We provide estimates for some moments of the solution, defined using the distance from a given vertex. Our technique relies on suitable inequalities of Faber-Krahn type, and looks at the local theory of continuous nonlinear partial differential equations. As it is known, however, not all of this approach can have a direct counterpart in graphs. A basic tool here is a result connecting the supremum of the solution at a given positive time with the measure of its level sets at previous times. We also consider the case of slowly decaying initial data, where the total mass is infinite

Asymptotic properties of solutions to the Cauchy problem for degenerate parabolic equations with inhomogeneous density on manifolds

We consider the Cauchy problem for doubly nonlinear degenerate parabolic equations with inhomogeneous density on noncompact Riemannian manifolds. We give a qualitative classification of the behavior of the solutions of the problem depending on the behavior of the density function at infinity and the geometry of the manifold, which is described in terms of its isoperimetric function. We establish for the solutions properties as: stabilization of the solution to zero for large times, finite speed of propagation, universal bounds of the solution, blow up of the interface. Each one of these behaviors of course takes place in a suitable range of parameters, whose definition involves a universal geometrical characteristic function, depending both on the geometry of the manifold and on the asymptotics of the density at infinity

Universal Bounds at the Blow-Up Time for Nonlinear Parabolic Equations

We prove a priori supremum bounds for solutions to doubly degenerate nonlinear parabolic equations, with a forcing term f(x)u^p where u is the solution, p> 1, f is strongly dependent on the space variable x, as t approaches the time when u becomes unbounded. Such bounds are universal in the sense that they do not depend on u. Here f may become unbounded, or vanish, as x goes to 0. When f =1, we also prove a bound below, as well as uniform localization of the support, for subsolutions to the corresponding Cauchy problem

The Cauchy--Dirichlet Problem for the Porous Media Equation in Cone-Like Domains

We investigate the behavior for large times of nonnegative solutions to the Dirichlet problem in cone-like domains for the porous media equation. We obtain optimal estimates for the sup norm of the solution and for the size of its support. We also consider the case where a damping term depending on the space gradient of the solution appears. In this case we also identify the critical behavior of the damping term discriminating between decay to zero of a suitable moment of the solution as t → + ∞, and stabilization of the moment to a positive constant. © 2014 Society for Industrial and Applied Mathematics