相关论文: Well-posedness and regularity of solutions to neur…

Diffuse domain methods (DDMs) have garnered significant attention for approximating solutions to partial differential equations on complex geometries. These methods implicitly represent the geometry by replacing the sharp boundary interface…

We study the regularity and uniqueness of weak solutions of a degenerate parabolic equation, arising as the limit of a stochastic lattice model of self-propelled particles. The angle-average of the solution appears as a coefficient in the…

We present a method of solving a nonlinear Dirichlet problem with discontinuous boundary data and we give a probabilistic representation of the solution using the nonlocal branching process associated with the nonlinear term of the…

We prove continuity for bounded weak solutions of a nonlinear nonlocal parabolic type equation associated to a Dirichlet form with a rough kernel. The equation is allowed to be singular at the level zero, and solutions may change sign. If…

A system of drift-diffusion equations with electric field under Dirichlet boundary conditions is analyzed. The system of strongly coupled parabolic equations for particle density and spin density vector describes the spin-polarized…

The paper deals with second order parabolic equations on bounded domains with Dirichlet conditions in arbitrary Euclidean spaces. Their interest comes from being models for describing reaction-diffusion processes in several frameworks. A…

The present paper deals with the Cauchy problem of a compressible generic two-fluid model with capillarity effects in any dimension $N\geq2$. We first study the unique global solvability of the model in spaces with critical regularity…

We study a nonlocal wave equation with logarithmic damping which is rather weak in the low frequency zone as compared with frequently studied strong damping case. We consider the Cauchy problem for this model in the whole space and we study…

We consider a degenerate wave equation in one dimension, with drift and in presence of a leading operator which is not in divergence form. We impose a homogeneous Dirichlet boundary condition where the degeneracy occurs and a boundary…

We prove the existence of a global solution to the Cauchy problem for a nonlinear reaction-diffusion system coupled with a system of ordinary differential equations. The system models the propagation of a combustion front in a porous medium…

It is studied the Cauchy problem for the equations of Burgers' type but with bounded dissipation flux. Such equations degenerate to hyperbolic ones as the velocity gradient tends to infinity. Thus the discontinuous solutions are permitted.…

In this paper we establish the optimal regularity estimates for the Cauchy problem of stochastic kinetic equations with random coefficients in anisotropic Besov spaces. As applications, we study the nonlinear filtering problem for a…

We consider the Cauchy problem for the generalized Fornberg-Whitham equation with dissipation. This is one of the nonlinear, nonlocal and dispersive-dissipative equations. The main topic of this paper is an asymptotic analysis for the…

We study the Cauchy problem for the defocusing modified Korteweg-de Vries (mKdV) equation with step-like initial data approaching nonzero constants $c_l$ and $c_r$ as $x \to -\infty$ and $x\to+\infty$, respectively. Assuming $c_l>c_r>0$,…

An asymptotic expansion with respect to a small parameter of the solution of the Cauchy problem is constructed for a system of three transfer equations, two of which are singularly perturbed by the degeneracy of the entire senior part of…

The diffusion equation is a universal and standard textbook model for partial differential equations (PDEs). In this work, we revisit its solutions, seeking, in particular, self-similar profiles. This problem connects to the classical…

Our focus is on the fast diffusion equation driven by the $p$-Laplacian operator, that is $\partial_t u=\Delta_p u$ with $1<p<2$, posed in the whole space $\mathbb{R}^N$, $N\geq 2$. The nonnegative solutions are expected to converge in time…

Effective representation of 2D images is fundamental in digital image processing, where traditional methods like raster and vector graphics struggle with sharpness and textural complexity respectively. Current neural fields offer…

In this paper, we consider the Cauchy problem for semi-linear wave equations with structural damping term $\nu (-\Delta)^2 u_t$, where $\nu >0$ is a constant. As being mentioned in [8,10], the linear principal part brings both the diffusion…

In this work, we propose an innovative iterative direct sampling method to solve nonlinear elliptic inverse problems from a limited number of pairs of Cauchy data. It extends the original direct sampling method (DSM) by incorporating an…