Zeitschrift für angewandte Mathematik und Physik最新文献

This study focuses on the Cauchy problem associated with the two-component peakon system featuring a cubic nonlinearity, constrained to the class ((m,n)in C^{k}(mathbb {R}) cap W^{k,1}(mathbb {R})) with (kin mathbb {N}cup {0}). This system extends the celebrated Fokas–Olver–Rosenau–Qiao equation and the following nonlocal (two-place) counterpart proposed by Lou and Qiao:

where (m(t,x)=left( 1-partial _{x}^2right) u(t,x)). Employing an approach based on Lagrangian coordinates, we establish the local existence, uniqueness, and Lipschitz continuity of the data-to-solution map in the class (C^kcap W^{k,1}). Moreover, we derive criteria for blow-up of the local solution in this class.

In this paper, we prove the multiplicity and concentration behavior of complex-valued solutions for the following Kirchhoff–Schrödinger equation with magnetic field

where (varepsilon >0) is a small parameter, the nonlinearity f is involved in critical exponential growth in the sense of Trudinger–Moser inequality and both (V:mathbb {R}^{2}rightarrow mathbb {R}) and (A:mathbb {R}^{2}rightarrow mathbb {R}^{2}) are continuous potential and magnetic potential, respectively. Imposing a local constraint of potential V(x) first introduced from del Pino and Felmer, we get the multiplicity of solutions by way of the relationship between the number of the solutions and the topology of the set with V attaining the minimum. Our strategy of main proof is based on the variational methods combined with the penalization technique, the Trudinger–Moser inequality and Ljusternik–Schnirelmann theory, and our result is still new even without magnetic effect.

This work revisits a recent finding by the first author concerning the local convergence of a regularized scalar conservation law. We significantly improve the original statement by establishing a global convergence result within the Lebesgue spaces (L^infty _{textrm{loc}}(mathbb {R}^+;L^p(mathbb {R}))), for any (p in [1,infty )), as the regularization parameter (ell ) approaches zero. Notably, we demonstrate that this stability result is accompanied by a quantifiable rate of convergence. A key insight in our proof lies in the observation that the fluctuations of the solutions remain under control in low regularity spaces, allowing for a potential quantification of their behavior in the limit as (ell rightarrow 0). This is achieved through a careful asymptotic analysis of the perturbative terms in the regularized equation, which, in our view, constitutes a pivotal contribution to the core findings of this paper.

The long-time behaviors are an important topic in the study of reaction diffusion equations. It is of interest to understand effects of variable coefficients and degenerate diffusion coefficients on the dynamical properties of reaction diffusion equations. In this paper, we shall use some new methods and techniques to prove that the degradation of the diffusion coefficient of the prey and variable coefficients satisfying the appropriate conditions will not affect dynamical properties of the reaction–diffusion prey–predator model.

This paper concerns a diffusive predation model with nonlinear growth, cross-diffusion and protection zone terms. The main purpose is to investigate the effects of nonlinear growth and cross-diffusion on the coexistent solution when protection zone is present. Firstly, a priori estimate and the existence of positive solutions are discussed, including local and global existence. Then, some asymptotic properties of coexistent solutions induced by the mortality rate, nonlinear growth of predator and cross-diffusion are analyzed. It is revealed that there exist critical values related to certain principal eigenvalues such that the nonlinear growth, cross-diffusion and protection zone all have significant effects on the coexistent solutions; as far as the nonlinear growth concerned, we find that it has important influences on the coexistence region of two species undoubtedly. Biologically, this implies that these critical values greatly affect the survival of species.

In this research, we examine a diffusive predator–prey model with spatial memory. We begin by checking that the suggested model has a unique solution that is boundedness. The stability of each equilibrium is then examined. Local and global stability as well as bifurcations are investigated in the non-delayed model at stationary equilibrium. Then, we investigate the Hopf bifurcation using the delay as the bifurcation parameter. In order to back up our theoretical findings, we then give some numerical simulations.

It is known that Lamb waves in homogeneous traction-free plates can propagate with arbitrary phase velocity, spanning the admissible speed interval (0;(+infty )). However, as the current research shows, Lamb waves propagating in two-layered traction-free plates may have ‘forbidden’ phase velocities, at which no Lamb waves can propagate. The analysis is based on the approach comprising Cauchy complex formalism and the exponential fundamental matrix method.

In this paper, we deal with the global dynamics of a Lotka-Volterra competition-diffusion-advection system with nonlinear boundary conditions, including the existence, nonexistence and global stability of coexistence steady states. We start with the investigation of the principal eigenvalue of linearized system to get the local stability of steady states and then discuss the global dynamics in terms of competition coefficients.

We introduce a vorticity Leray-(alpha ) model with eddy viscosity depending on (d(x,partial Omega )^eta ) where (partial Omega ) is the boundary of the domain and (eta in ]0;1[). We prove that this system admits fairly regular weak solutions converging when (alpha ) goes to 0 to the solution of a reference system