Nonlinear Analysis-Real World Applications最新文献

In this paper we study the zero-viscosity limit of 2-D Boussinesq equations with vertical viscosity and zero diffusivity, which is a nonlinear system with partial dissipation arising in atmospheric sciences and oceanic circulation. The domain is taken as $R_{+}^2$ with Navier-type boundary. We prove the nonlinear stability of the approximate solution constructed by boundary layer expansion in conormal Sobolev space. The expansion order and convergence rates for the inviscid limit are also identified in this paper. Our paper extends a partial zero-dissipation limit result of Boussinesq system with full dissipation by Chae D. (2006) in the whole space to the case with partial dissipation and Navier boundary in the half plane.

This paper is concerned with the existence of nontrivial solutions for affine $p$-Laplace equations involving subcritical nonlinearities behaving at $u=0$ as $u^q$ with $q<p-1$ and at the infinity as $u^r$ with $r>p-1$. Since local Palais–Smale compactness for affine energy type functionals is an open hard question, the problem is overcome by means of a perturbative approach using the space norm. Mountain-pass critical points are constructed from a limit process of corresponding ones in the modified affine context. Compactness and stability of MP solution sets are also addressed.

In this paper we study $\infty$-Laplacian type diffusion equations in weighted graphs obtained as limit as $p\to \infty$ to two types of $p$-Laplacian evolution equations in such graphs. We propose these diffusion equations, that are governed by the subdifferential of a convex energy functionals associated to the indicator function of the set $K_{\infty }^G≔\left(u\in L^2(V,{\nu }_G):|u\left(y\right)-u\left(x\right)|\le 1ifx\sim y\right)$ and the set $K_{\infty }^w≔\left(u\in L^2(V,{\nu }_G):|u\left(y\right)-u\left(x\right)|\le \frac{1}{\sqrt{w_{xy}}}ifx\sim y\right)$ as models for sandpile growth in weighted graphs. Moreover, we also analyse the collapse of the initial condition when it does not belong to the stable sets $K_{\infty }^G$ or $K_{\infty }^w$ by means of an abstract result given in Bénilan (2003). We give an interpretation of the limit problems in terms of Monge–Kantorovich mass transport theory. Finally, we give some explicit solutions of simple examples that illustrate the dynamics of the sandpile growing or collapsing.

This paper is concerned with the global existence and large time behavior of classical solutions away from vacuum to the Cauchy problem of the 1D compressible quantum Navier–Stokes–Poisson equations with large initial perturbation. Moreover, we obtain the global strong/classical solution of Navier–Stokes–Poisson equations through the vanishing dispersion limit with certain convergence rates. We focus on the case that the viscosity depends on density linearly which extends the former results of constant viscosity in Zhang et al. (2022) by the second author. Some useful estimates are developed to deduce the uniform-in-time lower and upper bounds on the specific volume and the electric potential.

In this paper, we study the Cauchy problem of the drift–diffusion system arising from semiconductor model. We prove that if a certain nonlinear function of the initial data is small enough, in a Besov type space, then there is a global solution to this drift–diffusion system. We also provide an example of initial data satisfying that nonlinear smallness condition, but whose norm be chosen arbitrarily large in ${\dot{B}}_{\infty ,\infty }^{-2}\left(R^d\right)$.

The paper focuses on the pattern formation of a chemotactic cell aggregation model with a mechanism that density suppresses motility. The model exhibits four types of cell aggregation patterns: single-point peaks, hot spots, cold spots, and stripes, depending on the parameters and mean density. The analysis is performed in two ways. First, traditional instability analysis reveals the existence of two critical densities. This local analysis shows patterns emerge if the initial mean density lies between the two values. Second, a phase separation method using van der Waals’ double well potential reveals that pattern formation is possible in a bigger parameter regime that includes the one identified by the local analysis. This non-local analysis shows that pattern formation occurs beyond the parameter regimes of the classical local instability analysis.

This paper considers the stabilization problem of the von Kármán beam equation with a combined boundary control of nonlinear delays and nonlinear non-delays. The combined boundary controls are applied at the transverse and longitudinal boundaries of the von Kármán beam, respectively. In this paper the nonlinear semigroup method is adopted in the investigation for the establishment of the well-posedness of the resulting closed-loop system. Constructing an appropriate energy-like function, the exponential decay rate of energy of the closed-loop system is demonstrated by a generalized Gronwall-type integral inequality and the integral multiplier technique.

We study the disease-spreading dynamics of the West Nile virus (WNv) epidemic model under shifting climatic conditions. A WNv epidemic model is developed incorporating a shifting net growth term to depict the evolving mosquito habitat. First, we comprehensively characterize the spreading dynamics of mosquitoes for any given climate change speed compared with the intrinsic spreading speed of mosquitoes. Utilizing the results from mosquito dynamics, we determine the spreading dynamics of infected birds and mosquitoes, taking into account relationships among the shifting speed and the spreading speeds of mosquito and WNv. Ultimately, we find that infected mosquitoes and birds propagate, and their population densities converge to a stable positive endemic state. This paper provides crucial insights into the impact of climate change on the spread of vector-borne diseases such as WNv.

We investigate the existence of weak solutions to a multi-component system, consisting of compressible chemically reacting components, coupled with the compressible Stokes equation for the velocity. Specifically, we consider the case of irreversible chemical reactions and assume a nonlinear relation between the pressure and the particular densities. These assumptions cause the additional difficulties in the mathematical analysis, due to the possible presence of vacuum.

It is shown that there exists a global weak solution, satisfying the $L^{\infty }$ bounds for all the components. We obtain strong compactness of the sequence of densities in $L^p$ spaces, under the assumption that all components are strictly positive. The applied method captures the properties of models of high generality, which admit an arbitrary number of components. Furthermore, the framework that we develop can handle models that contain both diffusing and non-diffusing elements.