Pointwise gradient estimates of solutions to onedimensional nonlinear parabolic equations

We present here an improved version of the method introduced by the first author to derive pointwise gradient estimates for the solutions of one-dimensional parabolic problems. After considering a general qualinear equation in divergence form we apply the method to the case of a nonlinear diffusion-convection equation. The conclusions are stated first for classical solutions and then for generalized and mild solutions. In the case of unbounded initial datum we obtain several regularizing effects for t > 0. Some unilateral pointwise gradient estimates are also obtained. The case of the Dirichlet problem is also considered. Finally, we collect, in the last section, several comments showing the connections among these estimates and the study of the free boundaries associated to the solutions of the diffusion-convection equation.     

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.     

Pointwise a posteriori error estimates for monotone semi-linear equations

We derive upper and lower a posteriori estimates for the maximum norm error in finite element solutions of monotone semi-linear equations. The estimates hold for Lagrange elements of any fixed order, non-smooth nonlinearities, and take numerical integration into account. The proof hinges on constructing continuous barrier functions by correcting the discrete solution appropriately, and then applying the continuous maximum principle; no geometric mesh constraints are thus required. Numerical experiments illustrate reliability and efficiency properties of the corresponding estimators and investigate the performance of the resulting adaptive algorithms in terms of the polynomial order and quadrature.     

Kernel estimates for a class of Kolmogorov semigroups

We prove short time pointwise upper bounds for the heat kernels of certain Kolmogorov operators. We use Lyapunov function techniques, where the Lyapunov functions depend also on the time variable. Received: 29 November 2007     

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.     

Marcinkiewicz estimates for degenerate parabolic equations with measure data

We consider solutions to degenerate parabolic equations with measurable coefficients, having on the right-hand side a measure satisfying a suitable density condition; we prove integrability results for the gradient in the Marcinkiewicz scale.     

Diffusive approximations for a class of nonlinear parabolic equations

We study a class of discrete velocity type approximations to nonlinear parabolic equations with source. After proving existence results and estimates on the solution to the relaxation system, we pass into the limit towards a weak solution, which is the unique entropy solution if the coefficients of the parabolic equation are constant.     

Error estimates for a mixed finite element discretization of some degenerate parabolic equations

We consider a numerical scheme for a class of degenerate parabolic equations, including both slow and fast diffusion cases. A particular example in this sense is the Richards equation modeling the flow in porous media. The numerical scheme is based on the mixed finite element method (MFEM) in space, and is of one step implicit in time. The lowest order Raviart–Thomas elements are used. Here we extend the results in Radu et al. (SIAM J Numer Anal 42:1452–1478, 2004), Schneid et al. (Numer Math 98:353–370, 2004) to a more general framework, by allowing for both types of degeneracies. We derive error estimates in terms of the discretization parameters and show the convergence of the scheme. The features of the MFEM, especially of the lowest order Raviart–Thomas elements, are now fully exploited in the proof of the convergence. The paper is concluded by numerical examples.     

Gradient estimates for Dirichlet parabolic problems in unbounded domains

We consider autonomous parabolic Dirichlet problems in a regular unbounded open set Ω⊂R^N involving second-order operator A with (possibly) unbounded coefficients. We determine new conditions on the coefficients of A yielding global gradient estimates for the bounded classical solution.     

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.     

Pullback attractors for a class of non-autonomous nonclassical diffusion equations

We consider the first initial boundary value problem for the non-autonomous nonclassical diffusion equation u_t−εΔu_t−Δu+f(u)=g(t), ε∈[0,1], in a bounded domain in R^N. Under a Sobolev growth rate of the nonlinearity f and a suitable exponential growth of the external force g, using the asymptotic a priori estimate method, we prove the existence of pullback D-attractors in the space and the upper semicontinuity of at ε=0.     

Harnack estimates for a quasi-linear parabolic equation with a singular weight

In this paper, based on measure theoretical arguments, we establish Harnack estimates and Hölder continuity of nonnegative weak solutions for a degenerate parabolic equation with a singular weight. We transform the equation by performing the change of function. The energy estimates, the upper boundedness, the lower boundedness and the expansion of positivity for the solutions to the transformed equation are obtained. Then our aim is reached.     

Sharp profiles in degenerate and doubly degenerate Fisher-KPP equations

This paper investigates the effects of a degenerate diffusion term in reaction-diffusion models u_t=[D(u)u_x]_x+g(u) with Fisher-KPP type g. Both in the case when D(0)=0 and when D(0)=D(1)=0, with D(u)>0 elsewhere, we obtain a continuum of travelling wave solutions having wave speed c greater than a threshold value c∗ and we show the appearance of a sharp-type profile when c=c∗. These results solve recent conjectures formulated by Sánchez-Garduño and Maini (J. Differential Equations 117 (1995) 281) and Satnoianu et al. (Discrete Continuous Dyn. Systems (Series B) 1 (2000) 339).     

Waveform relaxation for reaction-diffusion equations

We report a new waveform relaxation (WR) algorithm for general semi-linear reaction-diffusion equations. The superlinear rate of convergence of the new WR algorithm is proved, and we also show the advantages of the new approach superior to the classical WR algorithms by the estimation on iteration errors. The corresponding discrete WR algorithm for reaction-diffusion equations is presented, and further the parallelism of the discrete WR algorithm is analyzed. Moreover, the new approach is extended to handle the coupled reaction-diffusion equations. Numerical experiments are carried out to verify the effectiveness of the theoretic work.     

Hölder continuity of weak solutions of a class of linear equations with boundary degeneracy

This paper deals with a class of linear equations with boundary degeneracy. According to the degenerate ratio, the equations are divided into weakly degenerate ones and strongly degenerate ones, which should be supplemented by different Dirichlet boundary value conditions. After establishing some necessary existence, nonexistence and comparison principles, we investigate the optimal Hölder continuity of weak solutions in these two cases utilizing the Harnack inequality and the Morrey theorem, respectively.     

Quasilinear parabolic and elliptic equations with nonlinear boundary conditions

This paper is concerned with a class of quasilinear parabolic and elliptic equations in a bounded domain with both Dirichlet and nonlinear Neumann boundary conditions. The equation under consideration may be degenerate or singular depending on the property of the diffusion coefficient. The consideration of the class of equations is motivated by some heat-transfer problems where the heat capacity and thermal conductivity are both temperature dependent. The aim of the paper is to show the existence and uniqueness of a global time-dependent solution of the parabolic problem, existence of maximal and minimal steady-state solutions of the elliptic problem, including conditions for the uniqueness of a solution, and the asymptotic behavior of the time-dependent solution in relation to the steady-state solutions. Applications are given to some heat-transfer problems and an extended logistic reaction–diffusion equation.     

On a system of reaction-diffusion equations arising from competition with internal storage in an unstirred chemostat

In this paper we study a system of reaction-diffusion equations arising from competition of two microbial populations for a single-limited nutrient with internal storage in an unstirred chemostat. The conservation principle is used to reduce the dimension of the system by eliminating the equation for the nutrient. The reduced system (limiting system) generates a strongly monotone dynamical system in its feasible domain under a partial order. We construct suitable upper, lower solutions to establish the existence of positive steady-state solutions. Given the parameters of the reduced system, we answer the basic questions as to which species survives and which does not in the spatial environment and determine the global behaviors. The primary conclusion is that the survival of species depends on species's intrinsic biological characteristics, the external environment forces and the principal eigenvalues of some scalar partial differential equations. We also lift the dynamics of the limiting system to the full system.     

Fourth-order compact schemes with adaptive time step for monodomain reaction–diffusion equations

Multigrid applied to fourth-order compact schemes for monodomain reaction–diffusion equations in two dimensions has been developed. The scheme accounts for the anisotropy of the medium, allows for any cellular activation model to be used, and incorporates an adaptive time step algorithm. Numerical simulations show up to a 40% reduction in computational time for complex cellular models as compared to second-order schemes for the same solution error. These results point to high-order schemes as valid alternatives for the efficient solution of the cardiac electrophysiology problem when complex cellular activation models are used.