Solutions Of Fast Diffusion Equation Research Articles

Optimal boundary regularity for fast diffusion equations in bounded domains

:We prove optimal boundary regularity for bounded positive weak solutions of fast diffusion equations in smooth bounded domains. This solves a problem raised by Berryman and Holland in 1980 for these equations in the subcritical and critical regimes. Our proof of the a priori estimates uses a geometric type structure of the fast diffusion equations, where an important ingredient is an evolution equation for a curvature-like quantity.

Best matching Barenblatt profiles are delayed

The growth of the second moments of the solutions of fast diffusion equations is asymptotically governed by the behavior of self-similar solutions. However, at next order, there is a correction term which amounts to a delay depending on the nonlinearity and on a distance of the initial data to the set of self-similar Barenblatt solutions. This distance can be measured in terms of a relative entropy to the best matching Barenblatt profile. This best matching Barenblatt function determines a scale. In new variables based on this scale, which are given by a self-similar change of variables if and only if the initial datum is one of the Barenblatt profiles, the typical scale is monotone and has a limit. Coming back to original variables, the best matching Barenblatt profile is delayed compared to the self-similar solution with same initial second moment as the initial datum. Such a delay is a new phenomenon, which has to be taken into account for instance when fitting experimental data.

Cauchy problem and initial traces for fast diffusion equation with space-dependent source

We consider existence of initial traces of nonnegative solutions for fast diffusion equation with space-dependent source and the solvability of the Cauchy problem when the initial datum is merely a function in \({L_{{\rm loc}}^1(R^N)}\) or even a Radon measure.

The Continuous Dependence on the Nonlinearities of Solutions of Fast Diffusion Equations

AbstractIn this paper, we consider the Cauchy problemWe will prove that(i) for mc < m,m0 < 1, |u(x, t, m)–u(x, t, m0)| → 0 as m → m0 uniformly on every compact subset of ℝN × ℝ+, where ;(ii) there is a C* that explicitly depends on m such that

Gradient estimates for [formula omitted] on manifolds and some Liouville-type theorems

In this paper, we first prove a localized Hamilton-type gradient estimate for the positive solutions of Porous Media type equations: u t = Δ F ( u ) , with F ′ ( u ) > 0 , on a complete Riemannian manifold with Ricci curvature bounded from below. In the second part, we study Fast Diffusion Equation (FDE) and Porous Media Equation (PME): u t = Δ ( u p ) , p > 0 , and obtain localized Hamilton-type gradient estimates for FDE and PME in a larger range of p than that for Aronson–Bénilan estimate, Harnack inequalities and Cauchy problems in the literature. Applying the localized gradient estimates for FDE and PME, we prove some Liouville-type theorems for positive global solutions of FDE and PME on noncompact complete manifolds with nonnegative Ricci curvature, generalizing Yauʼs celebrated Liouville theorem for positive harmonic functions.

Hamilton’s gradient estimates and Liouville theorems for fast diffusion equations on noncompact Riemannian manifolds

Let M be a complete noncompact Riemannian manifold of dimension n. In this paper, we derive a local gradient estimate for positive solutions of fast diffusion equations ∂ t u = Δu α , 1 ― 2 n < α < 1 n on M x (―∞, 0]. We also obtain a theorem of Liouville type for positive solutions of the fast diffusion equation.

Self-similar singular solution of fast diffusion equation with gradient absorption terms

The self-similar singular solution of the fast diffusion equation with nonlinear gradient absorption terms are studied. By a self-similar transformation, the self-similar solutions satisfy a boundary value problem of nonlinear ordinary differential equation (ODE). Using the shooting arguments, the existence and uniqueness of the solution to the initial data problem of the nonlinear ODE are investigated, and the solutions are classified by the region of the initial data. The necessary and sufficient condition for the existence and uniqueness of self-similar very singular solutions is obtained by investigation of the classification of the solutions. In case of existence, the self-similar singular solution is very singular solution.

Potential nonclassical symmetries and solutions of fast diffusion equation

Abstract The fast diffusion equation u t = ( u −1 u x ) x is investigated from the symmetry point of view in development of the paper by Gandarias [M.L. Gandarias, Phys. Lett. A 286 (2001) 153]. After studying equivalence of nonclassical symmetries with respect to a transformation group, we completely classify the nonclassical symmetries of the corresponding potential equation. As a result, new wide classes of potential nonclassical symmetries of the fast diffusion equation are obtained. The set of known exact non-Lie solutions are supplemented with the similar ones. It is shown that all known non-Lie solutions of the fast diffusion equation are exhausted by ones which can be constructed in a regular way with the above potential nonclassical symmetries. Connection between classes of nonclassical and potential nonclassical symmetries of the fast diffusion equation is found.

Universal Bounds for Global Solutions of Fast Diffusion Equations with Source

We prove a universal bound, independent of initial data, for all global nonnegative solutions of fast diffusion equations of the form , with zero Dirichlet boundary conditions. Up to now, such results were only available in the semilinear and slow diffusion cases m≥ 1.

Darcy's Law and the Theory of Shrinking Solutions of Fast Diffusion Equations

The behavior of the solutions of the degenerate parabolic equation \[ v_{t}=v\,v_{xx}+{\kappa}|v_x|^{2}, \qquad {\kappa}\in {\mathbb{R}}, \] a basic model in the theory of flows in porous media, depends strongly on the parameter ${\kappa}$. We show here a striking example of that variability in the case of compactly supported solutions having freeboundaries. We consider the initial-value problem with continuous and compactly supported initial data $v(x,0)=v_0(x)\ge 0$. When ${\kappa}>0$ it is well known that this problem admits a unique weak solution which is compactly supported and the following free-boundary conditions are satisfied: \ $ v=0,$ \ $ v_t={\kappa}\,|v_x|^2.$ The latter relation is a form of Darcy's law and determines a unique solution, hence a unique choice of the interface. We prove here that for ${\kappa}\le 0$ there exist infinitely many solutions $v\ge 0$ with the same initial data and an interface on which Darcy's law holds. Actually, the interfaces can be chosen as arbitrary Lipschitz continuous curves as long as the support shrinks. Therefore, Darcy's law does not play a selecting role for this free-boundary problem.