Zeitschrift für angewandte Mathematik und Physik最新文献

Undercompressive shocks and their role in solving Riemann problem are studied. Solutions to a special system of two hyperbolic equations representing conservation laws are investigated. On the one hand, this system of equations makes it possible to demonstrate the nonstandard solutions to the Riemann problem; on the other hand, this system of equations describes longitudinal–torsional waves in elastic rods. We use the traveling wave criterion for admissibility of shocks as the additional jump condition. If the dissipation parameters included in each of the equations of the system are different, then there are undercompressed waves.

This paper deals with the following chemotaxis system:

under homogeneous Neumann boundary conditions in a bounded domain ( Omega subset {mathbb {R}}^{n}, nge 2,) with smooth boundary. Here, the positive function (gamma in C ^{2}([0, +infty )) ) satisfies (gamma '(s)<0) and ( gamma ''(s)ge 0) for all (sge 0,) also (xi (s)= -(1-alpha ),gamma '(s) ) with (alpha in (0, 1)). For the above system, we prove that the corresponding initial boundary value problem admits a unique global classical solution which is uniformly in time bounded. This result is obtained for small initial data without any restriction on (mu .) The obtained result improves a recent result by Li and Lu (J Math Anal Appl 521:126902, 2023), which asserts the global existence of bounded classical solutions, provided that ( frac{(gamma '(s))^{2}}{gamma ''(s)} le frac{n}{2(n+1)^{3}}) and some conditions on initial data and (mu .) We should mention that in the special cases (gamma (s)=(1+s)^{-k},(k>0)) and (gamma (s)=textit{e}^{-chi s}, (chi >0),) the result in Li and Lu (2023) is obtained under conditions on k and (chi .) But, our result is without any restriction on k and (chi .)

We consider the following fractional Schrödinger equation involving critical exponent:

where (2_s^*=frac{2N}{N-2s}), ((y',y'') in mathbb {R}^{2} times mathbb {R}^{N-2}) and (V(y) = V(|y'|,y'')) and (Q(y) = Q(|y'|,y'')) are bounded nonnegative functions in (mathbb {R}^{+} times mathbb {R}^{N-2}). By using finite-dimensional reduction method and local Pohozaev-type identities, we show that if (frac{2+N-sqrt{N^2+4}}{4}< s <min {frac{N}{4}, 1}) and (Q(r,y'')) has a stable critical point ((r_0,y_0'')) with (r_0>0,; Q(r_0,y_0'') > 0) and ( V(r_0,y_0'') > 0), then the above problem has infinitely many solutions, whose energy can be arbitrarily large.

This paper deals with the semilinear Rayleigh–Stokes equation with the fractional derivative in time of order (alpha in (0,1)), which can be used to model anomalous diffusion in viscoelastic fluids. An operator family related to this problem is defined, and its regularity properties are investigated. We firstly give the concept of the mild solutions in terms of the operator family and then obtain the existence of global mild solutions by means of fixed point technique. Moreover, the existence and regularity of classical solutions are given.

We obtain threshold results for the existence, non-existence and multiplicity of normalized solutions for semi-linear elliptic equations in the exterior of a ball. To the best of our knowledge, it is the first result in the literature addressing this problem for the (L^2) supercritical case. In particular, we show that the prescribed mass can affect the number of normalized solutions and has a stabilizing effect in the mass supercritical case. Furthermore, in the threshold we find a new exponent (p = 6) when (N = 2), which does not seem to have played a role for this equation in the past. Moreover, our findings are “quite surprising” and completely different from the results obtained on the entire space and on balls. We will also show that the nature of the domain is crucial for the existence and stability of standing waves. As a foretaste, it is well-known that in the supercritical case these waves are unstable in (mathbb {R}^N.) In this paper, we will show that in the exterior domain they are strongly stable.

In this paper, we mainly consider global well-posedness and long time behavior of compressible Navier–Stokes equations without heat conduction in (L^p)-framework. This is a generalization of Peng and Zhai (SIMA 55(2):1439–1463, 2023), where they obtained the corresponding result in (L^2)-framework. Based on the key observation that we can release the regularity of non-dissipative entropy S in high frequency in Peng and Zhai (2023), we ultimately achieve the desired (L^p) estimate in the high frequency via complicated calculations on the nonlinear terms. In addition, we get the (L^p)-decay rate of the solution.

In this paper, we provide sharp criteria of global attraction for a class of non-autonomous reaction–diffusion equations with delay and Neumann conditions. Our methodology is based on a subtle combination of some dynamical system tools and the maximum principle for parabolic equations. It is worth mentioning that our results are achieved under very weak and verifiable conditions. We apply our results to a wide variety of classical models, including the non-autonomous variants of Nicholson’s equation or the Mackey–Glass model. In some cases, our technique gives the optimal conditions for the global attraction.

This article presents some controllability and stabilization results for a system of two coupled linear Schrödinger equations in the one-dimensional case where the state components are interacting through the Kirchhoff boundary conditions. Considering the system in a bounded domain, the null boundary controllability result is shown. The result is achieved thanks to a new Carleman estimate, which ensures a boundary observation. Additionally, this boundary observation together with some trace estimates, helps us to use the Gramian approach, with a suitable choice of feedback law, to prove that the system under consideration decays exponentially to zero at least as fast as the function (e^{-2omega t}) for some (omega >0).

This article is the second part of a previous article devoted to the deterministic aspects. Here, we present a comprehensive study on the development and application of a novel stochastic second-gradient continuum model for particle-based materials. An application is presented concerning colloidal crystals. Since we are dealing with particle-based materials, factors such as the topology of contacts, particle sizes, shapes, and geometric structure are not considered. The mechanical properties of the introduced second-gradient continuum are modeled as random fields to account for uncertainties. The stochastic computational model is based on a mixed finite element (FE), and the Monte Carlo (MC) numerical simulation method is used as a stochastic solver. Finally, the resulting stochastic second-gradient model is applied to analyze colloidal crystals, which have wide-ranging applications. The simulations show the effects of second-order gradient on the mechanical response of a colloidal crystal under axial load, for which there could be significant fluctuations in the displacements.

In this paper, we investigate the nonlocal generalized Sasa–Satsuma (ngSS) equation based on an improved Riemann–Hilbert method (RHM). Different from the traditional RHM, the t-part of the Lax pair plays a more important role rather than the x-part in analyzing the spectral problems. So we start from the t-part of the spectral problems. In the process of dealing with the symmetry reductions, we are surprised to find that the computation is much less than the traditional RHM. We can more easily derive the compact expression of N-soliton solution of the ngSS equation under the reflectionless condition. In addition, the general high-order N-soliton solution of the ngSS equation is also deduced by means of the perturbed terms and limiting techniques. We not only demonstrate different cases for the dynamics of these solutions in detail in theory, but also exhibit the remarkable features of solitons and breathers graphically by demonstrating their 3D, projection profiles and wave propagations. Our results should be significant to understand the nonlocal nonlinear phenomena and provide a foundation for fostering more innovative research that advances the theory.