The length-preserving elastic flow with free boundary on hypersurfaces in $$\mathbb {R}^n$$

It is known that solutions of nonlocal dispersal evolution equations do not become smoother in space as time elapses. This lack of space regularity would cause a lot of difficulties in studying transition fronts in nonlocal equations. In the present paper, we establish some general criteria concerning space regularity of transition fronts in nonlocal dispersal evolution equations with a large class of nonlinearities, which allows the applicability of various techniques for reaction–diffusion equations to nonlocal equations, and hence serves as an initial and fundamental step for further studying various important qualitative properties of transition fronts such as stability, uniqueness and asymptotic speeds. We also prove the existence of continuously differentiable and increasing interface location functions, which give a better characterization of the propagation of transition fronts and are of great technical importance.

In this paper we use the comparison method to study the generation and metastability of patterns for a class of local and nonlocal nonlinear evolution equations. We show that for a typical initial datum, the pattern generated can last for a very long time, but is eventually destroyed.

Slip systems and flow patterns in viscoplastic metallic sheets with dislocations

Existence, uniqueness and stability of the stationary solution to a nonlocal evolution equation arising in population dispersal

Previous article Next article Regularity Properties of Flows Through Porous Media: A CounterexampleD. G. AronsonD. G. Aronsonhttps://doi.org/10.1137/0119027PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] D. G. Aronson, Regularity propeties of flows through porous media, SIAM J. Appl. Math., 17 (1969), 461–467 10.1137/0117045 MR0247303 (40:571) 0187.03401 LinkISIGoogle Scholar[2] A. M. Il'In, , A. S. Kalashnikov and , O. A. Oleinik, Second order linear equations of parabolic type, Russian Math. Surveys, 17 (1962), 1–143 10.1070/rm1962v017n03ABEH004115 CrossrefGoogle Scholar[3] A. S. Kalašnikov, Formation of singularities in solutions of the equation of nonstationary filtration, Z. Vyčisl. Mat. i Mat. Fiz., 7 (1967), 440–444 MR0211058 (35:1940) 0184.53201 Google Scholar[4] S. N. Kruzhkov, Results on the character of the regularity of solutions of parabolic equations and some of their applications, Math. Z., 6 (1969), 97–108 Google Scholar[5] O. A. Ladyženskaja, , V. A. Solonnikov and , N. N. Ural'ceva, Linear and quasilinear equations of parabolic type, Translated from the Russian by S. Smith. Translations of Mathematical Monographs, Vol. 23, American Mathematical Society, Providence, R.I., 1967xi+648 MR0241822 (39:3159b) 0174.15403 Google Scholar[6] O. A. Oleinik, , A. S. Kalashnikov and , Yui-Lin' Chzou, The Cauchy problem and boundary problems for equations of the type of non-stationary filtration, Izv. Akad. Nauk SSSR. Ser. Mat., 22 (1958), 667–704 MR0099834 (20:6271) Google Scholar[7] R. E. Pattle, Diffusion from an instantaneous point source with a concentration-dependent coefficient, Quart. J. Mech. Appl. 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The existence, uniqueness, and global exponential stability of traveling wave solutions of a class of nonlinear and nonlocal evolution equations are established. It is assumed that there are two stable equilibria so that a traveling wave is a solution that connects them. A basic assumption is the comparison principle: a smaller initial value produces a smaller solution. When applied to di↵erential equations or integro-di↵erential equations, the result recovers and/or complements a number of existing ones.

We study the long-term behavior of the solution to a nonlocal evolution equation which describes the limit of Ising spin systems with Glauber dynamics and Kac potentials on a bounded domain. We then propose an unconditionally stable, convergent finite difference scheme to this equation. We prove that the scheme is uniquely solvable, inherits the properties of the original equation, and that the numerical solution will approach the true solution in the $L^\infty$-norm.

In this paper, we main discuss an efficient numerical algorithm for the fourth‐order nonlocal evolution equation with a weakly singular kernel. The second‐order fractional convolution quadrature rule and L1 method are proposed to approximate the Riemann–Liouville (R‐L) fractional integral term and the temporal Caputo derivative, respectively. In order to obtain a fully discrete method, the compact difference scheme is used to discretize the second‐order and fourth‐order spatial derivative. Further, two new approaches are adopted for stability analysis, and then the optimal error estimates in the discrete ‐norm and ‐norm are obtained. At last, we give three test problems to illustrate the validity of the methods.

Approximation and comparison for motion by mean curvature with intersection points

Perturbation method for nonlocal impulsive evolution equations

This paper investigates the optimal control for a class of nonlocal fractional evolution equations of order gamma in (1,2) in Banach spaces. An adequate definition of α-mild solutions is obtained and the existence, uniqueness and continuous dependence of α-mild solutions for the presented control system are also established. The existence of optimal pairs of nonlocal fractional evolution systems is also demonstrated with a view on the construction of the Lagrange problem. Finally, an example is propounded for the presentation of optimal control.

In this paper, we investigate an initial boundary value problem for a nonlocal p-Laplacian evolution equation with an inner absorption and weighted linear nonlocal boundary and initial conditions. By using the modified comparison principle and the method of upper-lower solution, we find the influence of weighted function on determining blow-up or not of nonnegative solutions.

A hierarchy of nonlocal nonlinear evolution equations and [formula omitted]-dressing method

Partial-approximate controllability of nonlocal fractional evolution equations via approximating method

Sufficient conditions for boundary controllability of nonlocal impulsive neutral integrodifferential evolution equations are explored. In the first problem, the outcomes are proven with the help of Sadovskii's fixed point theorem collaborated with strongly continuous semigroup theory. In the second problem, we considered nonautonomous evolution equations, and the result is proven by employing Sadovskii's fixed point theorem in collaboration with the evolution system. Examples are included to demonstrate the theoretical results for both types of equations.