Positivity, local smoothing, and Harnack inequalities for very fast diffusion equations

We investigate qualitative properties of local solutions u(t,x)?0 to the fast diffusion equation, t_∂u=Δ(um)/m with m<1, corresponding to general nonnegative initial data. Our main results are quantitative positivity and boundedness estimates for locally defined solutions in domains of the form [0,T]×Ω, with Ω⊆Rd. They combine into forms of new Harnack inequalities that are typical of fast diffusion equations. Such results are new for low m in the so-called very fast diffusion range, precisely for all m?mc=(d−2)/d. The boundedness statements are true even for m?0, while the positivity ones cannot be true in that range.     

A multidimensional nonlinear sixth-order quantum diffusion equation

This paper is concerned with the analysis of a sixth-order nonlinear parabolic equation whose solutions describe the evolution of the particle density in a quantum fluid. We prove the global-in-time existence of weak nonnegative solutions in two and three space dimensions under periodic boundary conditions. Moreover, we show that these solutions are smooth and classical whenever the particle density is strictly positive, and we prove the long-time convergence to the spatial homogeneous equilibrium at a universal exponential rate. Our analysis strongly uses the Lyapunov property of the entropy functional.     

Convergence to steady states in a viscous Hamilton-Jacobi equation with degenerate diffusion

This paper deals with weak solutions of the one-dimensional viscous Hamilton-Jacobi equation

     

Nesting inertial manifolds for reaction and diffusion equations with large diffusivity

We study the asymptotic behaviour in large diffusivity of inertial manifolds governing the long time dynamics of a semilinear evolution system of reaction and diffusion equations. A priori, we review both local and global dynamics of the system in scales of Banach spaces of Hilbert type and we prove the existence of a universal compact attractor for the equations. Extensions yield the existence of a family of nesting inertial manifolds dependent on the diffusion of the system of equations. It is introduced an upper semicontinuity notion in large diffusivity for inertial manifolds. The limit inertial manifold whose dimension is strictly less than those of the infinite dimensional system of semilinear evolution equations is obtained.     

Blow up for nonlinear parabolic equations with time degeneracy under dynamical boundary conditions

We study the blow up behaviour of nonlinear parabolic equations including a time degeneracy, under dynamical boundary conditions. For some exponential and polynomial degeneracies, we develop some energy methods and some spectral comparison techniques and derive upper bounds for the blow up times.     

Existence of solutions of the very fast diffusion equation in bounded and unbounded domain

We prove the existence of a unique solution of the following Neumann problem , u > 0, in (a, b) × (0, T), u(x, 0) = u ₀(x) ≥ 0 in (a, b), and , where if m < 0, if m = 0, and m≤ 0, , and the case −1 < m ≤ 0, , for some constant p > 1 − m. We also obtain a similar result in higher dimensions. As a corollary we will give a new proof of a result of A. Rodriguez and J.L. Vazquez on the existence of infinitely many finite mass solutions of the above equation in for any −1 < m ≤ 0. We also obtain the exact decay rate of the solution at infinity.     

Asymptotic equipartition and long time behavior of solutions of a thin-film equation

We investigate the large-time behavior of classical solutions to the thin-film type equation ut=−x(uuxxx). It was shown in previous work of Carrillo and Toscani that for non-negative initial data u₀ that belongs to H¹(R) and also has a finite mass and second moment, the strong solutions relax in the L¹(R) norm at an explicit rate to the unique self-similar source type solution with the same mass. The equation itself is gradient flow for an energy functional that controls the H¹(R) norm, and so it is natural to expect that one should also have convergence in this norm. Carrillo and Toscani raised this question, but their methods, using a different Lyapunov functions that arises in the theory of the porous medium equation, do not directly address this since their Lyapunov functional does not involve derivatives of u. Here we show that the solutions do indeed converge in the H¹(R) norm at an explicit, but slow, rate. The key to establishing this convergence is an asymptotic equipartition of the excess energy. Roughly speaking, the energy functional whose dissipation drives the evolution through gradient flow consists of two parts: one involving derivatives of u, and one that does not. We show that these must decay at related rates—due to the asymptotic equipartition—and then use the results of Carrillo and Toscani to control the rate for the part that does not depend on derivatives. From this, one gets a rate on the dissipation for all of the excess energy.     

On some global well-posedness and asymptotic results for quasilinear parabolic equations

We prove that the quasilinear parabolic initial-boundary value problem (1.1) below is globally well-posed in a class of high order Sobolev solutions, and that these solutions possess compact, regular attractors ast+.     

Inertial manifolds and normal hyperbolicity

Our aim in this article is to derive an existence theorem of inertial manifolds for fairly general equations with a self-adjoint or nonself-adjoint linear operator in a Banach space setting. A sharp form of the spectral gap condition is given. Many other properties are proven including an interesting characterization of the inertial manifold and the normal hyperbolicity of the inertial manifold.     

Model of chemotaxis with threshold density and singular diffusion

A quasilinear singular parabolic system corresponding to recent models of chemotaxis in which (1) there is an impassable threshold for the density of cells and (2) the diffusion of cells becomes singular (fast or superdiffusion) when the density approaches the threshold. It is proved that for some range of parameters describing the relation between the diffusive and the chemotactic component of the cell flux there are global-in-time classical solutions which in some cases are separated from the threshold uniformly in time. Global-in-time weak solutions in the case of fast diffusion and the set of stationary states are studied as well. The applications of the general results to particular models are shown.     

Existence and regularity for a nonlinear boundary flow problem of population genetics

In this paper we consider a heat equation with nonlinear boundary condition occurring in population genetics, the selection–migration problem for alleles in a region, considering flow of genes throughout the boundary. Such a problem determines a gradient flow in a convenient phase space and then the dynamics for large times depends heavily on the knowledge of the equilibrium solutions. We address the questions of the existence of a nontrivial equilibrium solution and its regularity.     

A condition guaranteeing the absence of the blow-up phenomenon for the generalized Burgers equation

The initial boundary value problem for the generalized Burgers equation with nonlinear sources is considered. We formulate a condition guaranteeing the absence of the blow-up of a solution and discuss the optimality of this condition.     

A note on a fourth order degenerate parabolic equation in higher space dimensions

We are concerned with existence, positivity property and long-time behavior of solutions to the following initial boundary value problem of a fourth order degenerate parabolic equation in higher space dimensions      

Global nonexistence for a semilinear Petrovsky equation

In this paper we consider a semilinear Petrovsky equation with damping and source terms. It is proved that the solution blows up in finite time if the positive initial energy satisfies a suitable condition. Moreover for the linear damping case, we show that the solution blows up in finite time even for vanishing initial energy. This is an important breakthrough, since it is only well known that the solution blows up in finite time if the initial energy is negative from all the previous literature.     

Existence and uniqueness of solutions for a non-uniformly parabolic equation

In this paper, we study a generalized Burgers equation u_t+(u²)_x=tu_xx, which is a non-uniformly parabolic equation for t>0. We show the existence and uniqueness of classical solutions to the initial-value problem of the generalized Burgers equation with rough initial data belonging to .     

Local smoothing effects, positivity, and Harnack inequalities for the fast p-Laplacian equation

We study qualitative and quantitative properties of local weak solutions of the fast p-Laplacian equation, t_∂u=Δpu, with 1<p<2. Our main results are quantitative positivity and boundedness estimates for locally defined solutions in domains of Rn×[0,T]. We combine these lower and upper bounds in different forms of intrinsic Harnack inequalities, which are new in the very fast diffusion range, that is when 1<p?2n/(n+1). The boundedness results may be also extended to the limit case p=1, while the positivity estimates cannot.We prove the existence as well as sharp asymptotic estimates for the so-called large solutions for any 1<p<2, and point out their main properties.We also prove a new local energy inequality for suitable norms of the gradients of the solutions. As a consequence, we prove that bounded local weak solutions are indeed local strong solutions, more precisely .     

Dynamics near extinction time of a singular diffusion equation

We will show that if u is the solution of the equation , in is an even function on and is monotone decreasing in on , , where is a monotone increasing function satisfying with being given by and , then the rescaled function , will converge uniformly on every compact subset of to as where . Received: 25 May 2000 / Revised version: 26 October 2001 / Published online: 28 February 2002     

Numerical solution of a Cauchy problem for nonlinear reaction diffusion processes

In this paper, the authors propose a numerical method to compute the solution of the Cauchy problem: _wt-(w^m_wx)x=w^p$w_t-(w^m{w}_x{)}_x=w^p$, the initial condition is a nonnegative function with compact support, m>0$m>0$, p?m+1$p?m+1$. The problem is split into two parts: a hyperbolic term solved by using the Hopf and Lax formula and a parabolic term solved by a backward linearized Euler method in time and a finite element method in space. The convergence of the scheme is obtained. Further, it is proved that if m+1?p<m+3$m+1?p<m+3$, any numerical solution blows up in a finite time as the exact solution, while for p>m+3$p>m+3$, if the initial condition is sufficiently small, a global numerical solution exists, and if p?m+3$p?m+3$, for large initial condition, the solution is unbounded.     

Parabolic equations of Von Karman type on Kähler manifolds

We study a parabolic version of a system of Von Karman type on a compact Kähler manifold of arbitrary dimension. We provide local in time regular solutions, which can be extended to global bounded ones if the data of the problem are small.