Recent Progress on Reaction-Diffusion Systems and Viscosity Solutions

The following sections are included:

• PREFACE

• CONTENTS

The discovery of a priori bounds for semilinear parabolic equations with superlinear nonlinearity enables the use of the parabolic semiflow in proving existence results for equilibria, as an alternative to minimax methods from the calculus of variations. We present an overview of results that use this approach, discuss its shortcomings and advantages, and outline possible future research directions.

Reaction-diffusion systems with skew-gradient structure can be viewed as a sort of activator-inhibitor systems. We use variational methods to study the existence of steady state solutions. Furthermore, there is a close relation between the stability of a steady state and its relative Morse index. Some numerical results will also be disussed.

In this paper, we report some recent theoretical work of Du-Shi¹⁴ , Du-Liang¹² and Du-Peng-Wang¹³ that examines the effect of environmental changes in three well-known diffusive population models, each modeling the interaction of two population species in a common habitat. For each model, we consider the situation that one of the two species is endangered and a simple protection zone is created in the habitat for the endangered species. The focus is on the effect of the protection zone on the dynamics of the two species systems. We demonstrate that, though the effects of the protection zone on the dynamical behavior of the three systems are significantly different from each other, they all share one important property, namely, there exists a critical patch size for the protection zone: If the protection zone is below this size, each model behaves similarly to the non-protection zone case, but every model undergoes profound changes in dynamical behavior once the protection zone is above the critical patch size, and in such a case the endangered species is always saved from extinction.

The following sections are included:

• De Giorgi conjecture

• Available results

• Acknowledgments

• References

We show firstly the equivalence between existence of a periodic solution of the Hamilton-Jacobi equation u_t + H(x, Du) = f(t) in Ω × ℝ, where Ω is a bounded domain of ℝⁿ, with the Dirichlet boundary condition u = g(x,t) and that of a subsolution of the stationary problem H(x, Dv) = 〈f〉 under the assumptions that the function (f(t),g(x, t)) is periodic in t and H is coercive. Here 〈f〉 denotes the average of f over the period. This proposition is a variant of a recent result for Ω = ℝⁿ due to Bostan-Namah, and we give a different and simpler approach to such an equivalence. Secondly, we establish that any periodic solution u(x, t) of the problem, u_t + H(x, Du) = 0 in Ω × ℝ and u = g on ∂Ω × ℝ, is constant in t on the Aubry set for H. Here H is assumed to be convex, coercive and strictly convex in a sense.

We are concerned with an asymptotic expansion method which is applied to the volatility coefficient in the Black-Scholes models. Employing the partial differential equation (PDE) approach in a weak setting, we show that the explicit asymptotic pricing formula is possible together with mathematical convergence result.

The ABP type maximum principle for L^p-viscosity solutions of fully nonlinear second order elliptic/parabolic partial differential equations with unbounded coefficients and inhomogeneous terms is exhibited.

In this article, we will review various important topics in the heart modeling, especially in the electrophysiology aspect. The theme includes: modeling process for three different scales (cellular scale, tissue scale, whole heart scale), numerical methods, applications of computer simulation on heart arrhythmia, and challenges in the heart modeling.

This survey is concerned with positive solutions of nonlinear parabolic equations. Assuming that the underlying domain and the equation have certain reflectional symmetries, the presented results show how positive solutions reflect the symmetries. Depending on the class of solutions considered, the symmetries for all times or asymptotic symmetries are established. Several classes of problems, including fully nonlinear equations on bounded domains, quasi-linear equations on ℝ^N, asymptotically symmetric equations, and cooperative parabolic systems, are examined from this point of view. Applications of the symmetry results in the study of asymptotic temporal behavior of solutions are also shown.

The following sections are included:

• Introduction and main result

• Existence of travelling pulse waves

• Spectral property of pulse wave solutions

• References

A class of algorithms for computing high order geometric motions of planar curves is developed. The algorithms alternate between two simple steps—a convolution step and a "thresholding" step—to evolve planar curves according to combinations of Willmore flow, surface diffusion flow, and more general curvature dependent flows. When a closed curve is represented by a level set function and the thresholding step is replaced by redistancing the level set function according to its zero level contour, the proposed algorithm provide efficient and accurate computations of these geometric motions on uniform grids.

In this survey paper, we shall give a brief account on some aspects of blow-up problems for partial differential equations and systems of parabolic type with nonlinear boundary conditions, localized terms, nonlocal terms and problems not in divergence form. We are mainly concerned with blow-up rates, blowup sets, blow-up profiles, blow-up boundary layers, total blow-up, single point blow-up and simultaneous blow-up.

The following sections are included:

• Problems

• Global existence results

• A priori estimate

• Idea of proof of Theorem 2.6

• Idea of proof of Theorem 2.7

• Concluding remarks

• References

This article is concerned with the behavior of global solutions of a parabolic partial differential equation with a power nonlinearity. Depending on parameter values and initial data, various behavior of global solutions can be observed such as grow-up, convergence, quasi-convergence and non-convergence. We give precise descriptions of the relationship between the spatial decay rate of initial data and the large-time behavior of solutions.

We first give a brief review on traveling waves, spreading speeds, and global stability for monotone evolution systems with monostable and bistable nonlinearities. Then we outline our recently developed theory and methods for general monotone semiflows and certain non-monotone systems, and their applications to some biological models.