Large energy entire solutions for the Yamabe equation

We consider the Yamabe equation in Rn, n?3. Let k?1 and . For all large k we find a solution of the form , where , for n?4, for n=3 and o(1)→0 uniformly as k→+∞.     

Uniform decay estimates for finite-energy solutions of semi-linear elliptic inequalities and geometric applications

We prove uniform decay estimates at infinity for solutions 0?u∈Lp of the semilinear elliptic inequality Δu+auσ+bu?0, a,b?0, σ?1, in the presence of a Sobolev inequality (with potential term). This gives a unified point of view in the investigation of different geometric questions. In particular, we present applications to the study of the topology at infinity of parallel mean curvature submanifolds, to the non-compact Yamabe problem, and to estimate the decay rate of the traceless Ricci tensor of conformally flat manifolds.     

Isolated singularities of solutions to the Yamabe equation

In this paper we study the asymptotic behavior of local solutions to the Yamabe equation near an isolated singularity, when the metric is not necessarily conformally flat. We are able to prove, when the dimension is less than or equal to five, that any solution is asymptotic to a rotationally symmetric Fowler solution. We also prove refined asymptotics if deformed Fowler solutions are allowed in the expansion.     

Improved entropy decay estimates for the heat equation

Improved entropy decay estimates for the heat equation are obtained by selecting well-parametrized Gaussians. Either by mass centering or by fixing the second moments or the covariance matrix of the solution, relative entropy toward these Gaussians is shown to decay with better constants than classical estimates.     

On time decay estimates for solutions of the second mixed problem for a quasilinear parabolic equation

We obtain conditions for the existence of a global solution of the second mixed problem for a quasilinear parabolic equation in an unbounded domain. The estimates for the decay of a solution that depend on the geometry of the domain are established. Translated from Matematychni Metody ta Fizyko-Mekhanichni Polya, Vol. 51, No. 1, pp. 25–32, January–March, 2008.     

A note on the optimal temporal decay estimates of solutions to the Cahn-Hilliard equation

This paper is concerned with the optimal temporal decay estimates on the solutions of the Cauchy problem of the Cahn-Hilliard equation. It is shown in Liu, Wang and Zhao (2007) [11] that such a Cauchy problem admits a unique global smooth solution u(t,x) provided that the smooth nonlinear function φ(u) satisfies a local growth condition. Furthermore if φ(u) satisfies a somewhat stronger local growth condition, the optimal temporal decay estimates on u(t,x) are also obtained in Liu, Wang and Zhao (2007) [11]. Thus a natural question is how to deduce the optimal temporal decay estimates on u(t,x) only under the local growth condition which is sufficient to guarantee the global solvability of the corresponding Cauchy problem and the main purpose of this paper is devoted to this problem. Our analysis is motivated by the technique developed recently in Ukai, Yang and Zhao (2006) [15] with a slight modification.     

Uniform stability estimates for solutions and their gradients to the Boltzmann equation: A unified approach

As to the Cauchy problem for the spatially inhomogeneous Boltzmann equation with cut-off, we prove uniform stability estimates for solutions and their gradients in a unified and elementary way.     

Uniform estimates for solutions of nonlinear Klein-Gordon equations

Uniform estimates in $\text{H}{}_0{}^1\text{(Ω)}$ of global solutions to nonlinear Klein-Gordon equations of the form $\text{u}{}_{\mathrm{tt}}\text{? Δu + mu = g(u)}\text{in}\text{Ω, u = 0}\text{in}\text{?Ω}$, where Ω is an open subset of $\text{R}$^N, m > 0, and g satisfies some growth conditions are established.     

Gradient estimates and Harnack inequalities for a Yamabe-type parabolic equation under the Yamabe flow

In this paper, we derive a series of gradient estimates and Harnack inequalities for positive solutions of a Yamabe-type parabolic partial differential equation (△-?t)u=pu+qu~(a+1) under the Yamabe flow. Here p,q∈C~(2,1)(M~n×[0,T]) and a is a positive constant.     

Spatial decay estimates for the heat equation via the maximum principle

Spatial decay estimates for the solution of the heat equation, similar to those obtained by other authors using energy inequalities, are established through use of the maximum principle.

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Résumé A l'aide du principe du maximum, on obtient des évaluations pour la décroissance spatiale des solutions de l'équation de la chaleur. Ces résultats ressemblent à ceux d'autres auteurs, qui utilisaient des inégalités pour l'énergie.

     

Boundary decay estimates for solutions of fourth-order elliptic equations

We obtain integral boundary decay estimates for solutions of fourth-order elliptic equations on a bounded domain with regular boundary. We apply these estimates to obtain stability bounds for the corresponding eigenvalues under small perturbations of the boundary.     

Extinction and decay estimates of solutions for a polytropic filtration equation with the nonlocal source and interior absorption

This paper deals with the extinction properties of solutions for the homogeneous Dirichlet boundary value problem with the nonlocal source and interior absorption where m,λ,k,q > 0, 0 < m(p ? 1) < 1, r ≤ 1, and . By using L^p‐integral norm estimate method, we obtain the sufficient conditions of extinction solutions. Moreover, we also give the precise decay estimates of the extinction solutions. Copyright © 2012 John Wiley & Sons, Ltd.     

The decay mode solutions for the cylindrical KP equation

The decay mode solutions for the cylindrical Kadomtsev-Petviashvili equation can be obtained by the Bäcklund transformation and Hirota method.     

Maximal estimates for solutions to the nonelliptic Schrodinger equation

Maximal estimates are studied for solutions to an initial valueproblem for the nonelliptic Schrödinger equation. A resultof Rogers, Vargas and Vega is extended.     

Explicit decay estimates for solutions to nonlinear parabolic systems

The aim of this paper is to investigate a class of nonlinear parabolic systems with initial and boundary values of Dirichlet type, when the nonlinearities depend on the gradient of the solution. Sufficient conditions on data are established in order to preclude blow up and to deduce that the solution decays exponentially in time. Moreover, an upper bound of its gradient is derived.