A quenching criterion for a multi-dimensional parabolic problem due to a concentrated nonlinear source

A multi-dimensional parabolic first initial-boundary value problem with a concentrated nonlinear source is studied. A criterion for its solution to quench, in a finite time t_q, everywhere on the concentrated nonlinear source only is given. An upper bound for t_q is also deduced. For illustration, an example is given.     

Universal bounds for global solutions of a diffusion equation with a nonlocal reaction term

We consider the nonlinear heat equation with nonlocal reaction term in space , in smoothly bounded domains. We prove the existence of a universal bound for all nonnegative global solutions of this equation. Moreover, in contrast with similar recent results for equations with local reaction terms, this is shown to hold for all p>1. As an interesting by-product of our proof, we derive for this equation a smoothing effect under weaker assumptions than for corresponding problem with local reaction.     

Large time behavior of free boundary for a degenerate parabolic equation with absorption

This work studies the large time behavior of free boundary and continuous dependence on nonlinearity for the Cauchy problem of a degenerate parabolic partial differential equation with absorption. Our objective is to give an explicit expression of speed of propagation of the solution and to show that the solution depends on the nonlinearity of the equation continuously.     

Variational approach for a class of cooperative systems

The aim of this work is to ascertain the characterization of the existence of coexistence states for a class of cooperative systems supported by the study of an associated non-local equation through classical variational methods. Thanks to those results, we are able to obtain the blow-up behavior of the solutions in the whole domain for certain values of the main continuation parameter.     

Evolving a convex closed curve to another one via a length-preserving linear flow

Motivated by a recent curvature flow introduced by Professor S.-T. Yau [S.-T. Yau, Private communication on his “Curvature Difference Flow”, 2007], we use a simple curvature flow to evolve a convex closed curve to another one (under the assumption that both curves have the same length). We show that, under the evolution, the length is preserved and if the curvature is bounded above during the evolution, then an initial convex closed curve can be evolved to another given one.     

The instability of spiky steady states for a competing species model with cross diffusion

This paper is concerned with the stability/instability of a class of positive spiky steady states for a quasi-linear cross-diffusion system describing two-species competition. By detailed spectral analysis, it is proved that the spiky steady states for the related shadow system are linearly unstable and the spiky steady states for the original cross-diffusion system are non-linearly unstable.     

Pointwise estimates for a class of non-homogeneous Kolmogorov equations

We consider a class of ultraparabolic differential equations that satisfy the Hörmander’s hypoellipticity condition and we prove that the weak solutions to the equation with measurable coefficients are locally bounded functions. The method extends the Moser’s iteration procedure and has previously been employed in the case of operators verifying a further homogeneity assumption. Here we remove that assumption by proving some potential estimates and some ad hoc Sobolev type inequalities for solutions.     

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.     

Blow-up phenomena for a system of semilinear heat equations with nonlinear boundary flux

This paper deals with the blow-up for a system of semilinear r-Laplace heat equations with nonlinear boundary flux. It is shown that, under certain conditions on the nonlinearities and data, blow-up will occur at some finite time, and when blow-up does occur upper and lower bounds for the blow-up time are obtained.     

Blow-up phenomena for a semilinear heat equation with nonlinear boundary condition, II

This paper deals with the blow-up of the solution to a semilinear second-order parabolic equation with nonlinear boundary conditions. It is shown that under certain conditions on the nonlinearities and data, blow-up will occur at some finite time and when blow-up does occur upper and lower bounds for the blow-up time are obtained.     

Global attractor for a non-autonomous integro-differential equation in materials with memory

The long-time behavior of an integro-differential parabolic equation of diffusion type with memory terms, expressed by convolution integrals involving infinite delays and by a forcing term with bounded delay, is investigated in this paper. The assumptions imposed on the coefficients are weak in the sense that uniqueness of solutions of the corresponding initial value problems cannot be guaranteed. Then, it is proved that the model generates a multivalued non-autonomous dynamical system which possesses a pullback attractor. First, the analysis is carried out with an abstract parabolic equation. Then, the theory is applied to the particular integro-differential equation which is the objective of this paper. The general results obtained in the paper are also valid for other types of parabolic equations with memory.     

Asymptotic bounds of solutions for a periodic doubly degenerate parabolic equation

This paper is concerned with a doubly degenerate parabolic equation with logistic periodic sources. We are interested in the discussion of the asymptotic behavior of solutions of the initial-boundary value problem. In this paper, we first establish the existence of non-trivial nonnegative periodic solutions by a monotonicity method. Then by using the Moser iterative method, we obtain an a priori upper bound of the nonnegative periodic solutions, by means of which we show the existence of the maximum periodic solution and asymptotic bounds of the nonnegative solutions of the initial-boundary value problem. We also prove that the support of the non-trivial nonnegative periodic solution is independent of time.     

Singular and degenerate anisotropic parabolic equations with a nonlinear source

The Dirichlet problem in arbitrary domain for degenerate and singular anisotropic parabolic equations with a nonlinear source term is considered. We state conditions that guarantee the existence and uniqueness of a global weak solution to the problem. A similar result is proved for the parabolic p-Laplace equation.     

Uniqueness for a forward backward diffusion equation with smooth constitutive funcation

We show that the initial boundary value problem for the diffusion equation u_L ?(u_x)_x with bounded Neumann boundary conditions has at most one smooth solutions on the infinite cylinder provided the nonmonotonic function ? is piecewise continuously differentiable on the reals has at most finitely many extreme values on any bounded interval,satisfies the coercivity condition s?(s) ≤ cs², c < 0 and has a derivative which is nonzero almost everywhere. In particular this result provides uniquencess for the model cubic ?(s)=s³ /3-3s²/2+2s proposed by Hölling and Nohel     

Uniform blow-up rate for a nonlocal degenerate parabolic equations

In this paper, we introduce a new method for investigating the rate of blow-up of solutions of diffusion equations with nonlocal nonlinear reaction terms. In some cases, we prove that the solutions have global blow-up and the rate of blow-up is uniform in all compact subsets of the domain. In each case, the blow-up rate of _|u(t)|∞${\left|u\left(t\right)\right|}_{\infty }$ is precisely determined.     

Boundedness of global solutions of a porous medium equation with a localized source

In this paper a localized porous medium equation _ut=u^r(Δu+af(u(_x0,t)))$u_t=u^r(\mathrm{\Delta }u+\mathit{af}(u(x_0,t)))$ is considered. It is shown that under certain conditions solutions of the above equation blow up in finite time for large a or large initial data while there exist global positive solutions to the above equation for small a or small initial data. Moreover, it is also shown that all global positive solutions of the above equation are uniformly bounded, and this differs from that of a porous medium equation with a local source.     

Blow up rate for a porous medium equation with power nonlinearity

The present paper is concerned with the blow up rate of a solution to the following Cauchy problem

     

On a prey-predator reaction-diffusion system with Holling type III functional response

In this paper we study a prey-predator model defined by an initial-boundary value problem whose dynamics is described by a Holling type III functional response. We establish global existence and uniqueness of the strong solution. We prove that if the initial data are positive and satisfy a certain regularity condition, the solution of the problem is positive and bounded on the domain and then we deduce the continuous dependence on the initial data. A numerical approximation of the system is carried out with a spectral method coupled with the fourth-order Runge-Kutta time solver. The biological relevance of the comparative numerical results is also presented.     

Support properties of solutions to a degenerate equation with absorption and variable density

In this paper, we study the properties of solutions to a degenerate parabolic equation with variable density and absorption. We first obtain a critical exponent, which distinguishes the localization of solutions from the positivity of them. When positivity prevails, we obtain the other critical exponent with respect to the decay of the variable density, which separates the global existence of interfaces from the disappearance of them. Moreover, the long time behavior of interfaces is characterized.     

On a coupled nonlinear singular thermoelastic system

For a coupled nonlinear singular system of thermoelasticity with one space dimension, we consider its initial boundary value problem on an interval. For one of the unknowns a classical condition is replaced by a nonlocal constraint of integral type. Because of the presence of a memory term in one of the equations and the presence of a weighted boundary integral condition, the solution requires a delicate set of techniques. We first solve a particular case of the given nonlinear problem by using a functional analysis approach. On the basis of the results obtained and an iteration method we establish the well-posedness of solutions in weighted Sobolev spaces.