Studies in Applied Mathematics最新文献

Considered herein is a Cauchy problem for a system of Schrödinger equations with second-order derivative-type and combined power-type nonlinearities. Through the effective combination of potential well theory, conservation laws, and vector-valued Gargliardo–Nirenberg inequality, we establish the uniform boundedness in $�H$-norm on $��[�0�,�T�)�$ and corresponding decay rate estimate. Moreover, we also prove the existence of corresponding ground-state solutions for this problem. Finally, we mainly investigate three different sharp thresholds for blowup and uniform bound of solutions in $�H$-norm on $��[�0�,�T�)�$ by using potential well theory, variational method, and some transformation techniques.

We derive rigorously from the water waves equations new irrotational shallow-water models for the propagation of surface waves in the case of uneven topography in horizontal dimensions one and two. The systems are made to capture the possible change in the waves' propagation, which can occur in the case of large amplitude topography. The main contribution of this work is the construction of new multiscale shallow-water approximations of the Dirichlet–Neumann operator. We prove that the precision of these approximations is given at the order $��O�\left(�\mu �\epsilon �\right)�$, $��O�(�\mu �\epsilon �+�{\mu }^2�{\beta }^2�)�$, and $��O�(�{\mu }^2�\epsilon �+�\mu �\epsilon �\beta �+�{\mu }^2�{\beta }^2�)�$. Here, $�\mu$, $�\epsilon$, and $�\beta$ denote, respectively, the sh

We explore the relationship between rough surface conductors and the phenomenon of passive intermodulation. The underlying surface is taken to be the boundary of a Lipschitz domain, and a characteristic angle of the domain is used to track boundary dependence on the various fields. To model electro-thermal passive intermodulation in particular, we consider a specific type of temperature-dependent conductivity and determine conditions on the conductivity under which one can use fixed point arguments to solve an induction heating and Joule heating problem on a Lipschitz domain. In the latter problem, we also consider a time-dependent permittivity function $�\epsilon$. Finally, weak solutions to a magneto-quasi-static problem are obtained when the permeability µ is temperature dependent and is allowed to degenerate in a certain way. An interesting effect of the rough surface is the inherently limited Sobolev regularity of the electric field, which can be improved if one assumes additional constraints on the boundary.

We study global existence, boundedness, and convergence of nonnegative classical solutions to a Neumann initial-boundary value problem for the following possibly cross-diffusive SIS (susceptible–infected–susceptible) epidemic model with power-like infection mechanism generalizing the standard mass action incidence:

In this paper, we discuss the reduction of symplectic Hamiltonian systems by scaling and standard symmetries which commute. We prove that such a reduction process produces a so-called Kirillov Hamiltonian system. Moreover, we show that if we reduce first by the scaling symmetries and then by the standard ones or in the opposite order, we obtain equivalent Kirillov Hamiltonian systems. In the particular case when the configuration space of the symplectic Hamiltonian system is a Lie group $�G$, which coincides with the symmetry group, the reduced structure is an interesting Kirillov version of the Lie–Poisson structure on the dual space of the Lie algebra of $�G$. We also discuss a reconstruction process for symplectic Hamiltonian systems which admit a scaling symmetry. All the previous results are illustrated in detail with some interesting examples.

This paper is concerned with a free boundary problem with resource-dependent motility in a weak heterogeneous environment. The existence and uniqueness of global solutions are discussed first. Next, we establish long-time behaviors of solutions which is a spreading–vanishing dichotomy. Moreover, we obtain sharp criteria on spreading and vanishing by investigating the associated linearized eigenvalue problem. The theoretical analyses reveal an important biological phenomenon. (i) Resource abundance reduces the motility, providing advantages for individuals to persist in a habitat. (ii) Resource shortage enhances the motility, forcing individuals to expand outward to survive in an environment. Consequently, resource-dependent motility is more beneficial to the survival of species compared with random dispersal.

In this paper, a compound equation which is a mix of the Wadati–Konno–Ichikawa (WKI) equation and the short-pulse (SP) equation is first studied. By transforming both the independent and dependent variables in the equation, we introduce a novel hodograph transformation to convert the compound WKI–SP equation into the mKdV–SG (modified Korteweg–de Vries and sine-Gordon) equation. The multiloop soliton solutions in the form of the parametric representation are found. It is shown that the $�N$-loop soliton solution may be decomposed exactly into $�N$ separate soliton elements by using a Moloney–Hodnett-type decomposition. By virtue of the decomposed soliton solutions, the asymptotic behaviors of $��N�=�2�$ and $��N�=�3�$ are investigated in detail. The corresponding phase shifts of each loop or antiloop soliton caused by its interaction with the other ones are calculated. Furthermore, a new hierarchy of WKI–SP-type equations possessing multiloop soliton solutions is constructed. These deduced equations are all with time-varying coefficients and the corresponding dispersion relation will have a time-dependent velocity. The whole hierarchy of equations which include the WKI-type equations, the SP-type equations, and the compound generalized WKI–SP equations, are illustrated Lax integrable. The specific equation in the hierarchy is labeled as $��\mathrm{WKI}�\text{--}�{\mathrm{SP}}^{�(�n�,�m�)�}�$ equation so that its Lax pairs can be directly written out with the help of $�n$ and