Large time behavior of solutions to parabolic equations with Neumann boundary conditions

In this paper we are interested in the large time behavior as t→+∞ of the viscosity solutions of parabolic equations with nonlinear Neumann type boundary conditions in connection with ergodic boundary problems which have been recently studied by Barles and the author in [G. Barles, F. Da Lio, On the boundary ergodic problem for fully nonlinear equations in bounded domains with general nonlinear Neumann boundary conditions, Ann. Inst. H. Poincaré Anal. Non Linèaire 22 (5) (2005) 521-541].     

Global existence for quasilinear parabolic systems with homogeneous Neumann boundary conditions

We present a result on the global existence of classical solutions for quasilinear parabolic system in bounded domains with homogenous Neumann boundary conditions.     

Blow-up of a non-local semilinear parabolic equation with Neumann boundary conditions

In this paper we give a positive answer to the conjecture proposed in [A. El Soufi, M. Jazar, R. Monneau, A Gamma-convergence argument for the blow-up of a non-local semilinear parabolic equation with Neumann boundary conditions, Ann. Inst. H. Poincaré Anal. Non Linéaire 24 (1) (2007) 17–39] by El Soufi et al. concerning the finite time blow-up for solutions of the problem (1), (2) below. More precisely, we give a direct proof of [A. El Soufi, M. Jazar, R. Monneau, A Gamma-convergence argument for the blow-up of a non-local semilinear parabolic equation with Neumann boundary conditions, Ann. Inst. H. Poincaré Anal. Non Linéaire 24 (1) (2007) 17–39, Theorem 1.1] and the conjecture given for the case p>2$p>2$.     

Periodic solutions for a class of degenerate parabolic equations with Neumann boundary conditions

In this paper, we investigate a class of degenerate parabolic periodic equations with nonlocal terms under Neumann boundary conditions. By using the theory of the Leray–Schauder degree, we obtain the existence of non-trivial nonnegative periodic solutions.     

Blow-up and global solutions for quasilinear parabolic equations with Neumann boundary conditions

Using the upper and lower solution techniques and Hopf's maximum principle, the sufficient conditions for the existence of blow-up positive solution and global positive solution are obtained for a class of quasilinear parabolic equations subject to Neumann boundary conditions. An upper bound for the ‘blow-up time’, an upper estimate of the ‘blow-up rate’, and an upper estimate of the global solution are also specified.     

Coexistence periodic solutions of a doubly nonlinear parabolic system with Neumann boundary conditions

This paper is concerned with a competitive and cooperative mathematical model for two biological populations which dislike crowding, diffuse slowly and live in a common territory under different kind of intra- and inter-specific interferences. The model consists of a system of two doubly nonlinear parabolic equations with nonlocal terms and Neumann boundary conditions. Based on the theory of the Leray–Schauder degree, we obtain the coexistence periodic solutions, namely the existence of two non-trivial non-negative periodic solutions representing the densities of the two interacting populations, under different intra- and inter-specific interferences on their natural growth rates.     

Periodic solutions of the wave equation with nonconstant coefficients and with homogeneous Dirichlet and Neumann boundary conditions

We prove theorems on the existence and regularization of periodic solutions of the wave equation with variable coefficients on an interval with homogeneous Dirichlet and Neumann boundary conditions. The nonlinear term has a power-law growth or satisfies the nonresonance condition at infinity.     

Periodic solutions of a quasilinear wave equation with homogeneous boundary conditions

In this paper, we prove the existence of time-periodic weak solutions for the wave equation with homogeneous boundary conditions. This paper deals with the cases where a nonlinear term has a superlinear and sublinear growth. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 12, No. 5, pp. 189–201, 2005.     

Remarks on blow-up behavior for a nonlinear diffusion equation with Neumann boundary conditions

We establish the blow-up rate for the solution of a nonlinear diffusion equation , subject to Neumann boundary conditions .

     

Asymptotic behavior of viscosity solutions for a degenerate parabolic equation associated with the infinity-Laplacian

The asymptotic behavior of viscosity solutions to the Cauchy–Dirichlet problem for the degenerate parabolic equation u _t  = Δ_∞ u in Ω × (0,∞), where Δ_∞ stands for the so-called infinity-Laplacian, is studied in three cases: (i) and the initial data has a compact support; (ii) Ω is bounded and the boundary condition is zero; (iii) Ω is bounded and the boundary condition is non-zero. Our method of proof is based on the comparison principle and barrier function arguments. Explicit representations of separable type and self-similar type of solutions are also established. Moreover, in case (iii), we propose another type of barrier function deeply related to a solution of . Goro Akagi was supported by the Shibaura Institute of Technology grant for Project Research (no. 2006-211459, 2007-211455), and the grant-in-aid for young scientists (B) (no. 19740073), Ministry of Education, Culture, Sports, Science and Technology. Petri Juutinen was supported by the Academy of Finland project 108374. Ryuji Kajikiya was supported by the grant-in-aid for scientific research (C) (no. 16540179), Ministry of Education, Culture, Sports, Science and Technology.     

A Gamma-convergence argument for the blow-up of a non-local semilinear parabolic equation with Neumann boundary conditions

In this paper we study a simple non-local semilinear parabolic equation in a bounded domain with Neumann boundary conditions. We obtain a global existence result for initial data whose L^∞$L^{\infty }$-norm is less than a constant depending explicitly on the geometry of the domain. A natural energy is associated to the equation and we establish a relationship between the finite-time blow up of solutions and the negativity of their energy. The proof of this result is based on a Gamma-convergence technique.     

Asymptotic properties of singular solutions in degenerate parabolic equation with boundary flux

ABSTRACT

This paper deals with blow-up and quenching solutions of degenerate parabolic problem involving m-Laplacian operator and nonlinear boundary flux. The blow-up and quenching criteria are classified under the conditions on the initial data but with less conditions on the relationship among the exponents, respectively. Moreover, asymptotic properties including singular rates, set and time estimates are determined for the blow-up solutions and the quenching solutions, respectively.     

Global existence and large-time behavior of a parabolic phase-field model with Neumann boundary conditions

In this paper, we continue to study an initial boundary value problems to a model describing the evolution in time of diffusive phase interfaces in sea ice growth. In a previous paper, the global existence and the large-time behavior of weak solutions in one space was studied under Dirichlet boundary conditions. Here, we show that the global existence of weak solutions and the large-time behavior are also studied under Neumann boundary condition. In this paper, we study in space dimension lower than or equal to 3.     

On the blow-up behavior of a nonlinear parabolic equation with periodic boundary conditions

Using ideas arising in the works of LeJan and Sznitman and Mattingly and Sinai on their study of the Navier–Stokes equations, we investigate the blow-up behavior of a nonlinear parabolic equation subject to periodic boundary conditions.     

On a mixed problem with integral conditions for one-dimensional parabolic equation

We consider an initial-boundary value problem for a one-dimensional parabolic equation with nonlocal boundary conditions. These nonlocal conditions are given in terms of integrals. Based on solution of the Dirichlet problem for the parabolic equation, we constructively establish the well-posedness for the nonlocal problem.     

Asymptotic blow-up behavior for a nonlocal degenerate parabolic equation

In this paper, we investigate the positive solution of nonlinear degenerate equation with Dirichlet boundary condition. The blow-up criteria is obtained. Furthermore, we prove that under certain conditions, the solutions have global blow-up. When f(u)=u^p,0<p1, we gained blow-up rate estimate.     

Critical exponent for non-Newtonian filtration equation with homogeneous Neumann boundary data

This paper is concerned with large time behavior of solutions to the homogeneous Neumann problem of the non-Newtonian filtration equation. It is shown that the critical Fujita exponent for the problem considered is determined not only by the spatial dimension and the nonlinearity exponent, but also by the coefficient k of the first-order term. In fact, we show that there exist two thresholds k_∞ and k₁ on the coefficient k of the first-order term, and the critical Fujita exponent is a finite number when k is between k_∞ and k₁, while the critical exponent does not exist when k⩽k_∞ or k⩾k₁. Copyright © 2007 John Wiley & Sons, Ltd.