偏微分方程分析

In this paper, we first investigate quasi-entropy solutions to scalar conservation laws in several space dimensions. In this setting, we introduce a suitable Lagrangian representation for such solutions. Next, we prove that, in one space…

In this paper we establish gradient estimates for positive solutions to the nonlinear elliptic equation $$\Delta_{V}u^{m}+\mu(x)u+p(x)u^{\alpha}=0 , \quad m>1$$on any smooth metric measure space whose $k$-Bakry-\'{E}mery curvature is…

Existence, uniqueness and classification is established for plane waves supported by an irreversible reaction which is a smooth function of local reactant and product concentrations (or prey and predator populations). Rudimentary analytic…

For a fixed constant $\lambda > 0$ and a bounded Lipschitz domain $\Omega \subset \mathbb{R}^n$ with $n \geq 2$, we establish that almost-minimizers (functions satisfying a sort of variational inequality) of the Alt-Caffarelli type…

This study examines a fully parabolic predator-prey chemo-alarm-taxis system under homogeneous Neumann boundary conditions in a bounded domain $\Omega \subset \mathbb{R}^n$ with a smooth boundary $\partial\Omega$. Under specific parameter…

We analyze the trapping of diffusing ligands, modeled as Brownian particles, by a sphere that has $N$ partially reactive boundary patches, each of small area and arbitrary shape, on an otherwise reflecting boundary. For such a structured…

We study extrema of solutions to the heat equation (i.e. hot spots) on a class of warped product manifolds of the form $([0,L]\times M,dr^2+f(r)^2h)$ where $(M,h)$ is a closed Riemannian manifold. We prove that, under certain conditions on…

We consider weak solutions to the incompressible Euler equations. It is shown that energy conservation holds in any Onsager critical class in which smooth functions are dense. The argument is independent of the specific critical regularity…

This article considers the variational wave equation with viscosity and transport noise as a system of three coupled nonlinear stochastic partial differential equations. We prove pathwise global existence, uniqueness, and temporal…

The present contribution investigates the dynamics generated by the two-dimensional Vlasov-Poisson-Fokker-Planck equation for charged particles in a steady inhomogeneous background of opposite charges. We provide global in time estimates…

We consider a three-dimensional kinetic model for a two species plasma consisting of electrons and ions confined by an external nonconstant magnetic field. Then we derive a kinetic-fluid model when the mass ratio $m_e/m_i$ tends to zero.…

Here we explore, in a series of articles, semiclassical quasimodes u(h,b), approximative solutions P(h)u(h,b)\sim 0, depending on $0<h<1$, and on b, the subprincipal symbol. We study a pseudodifferential operator with transversal…

We construct self-similar solutions to the 2D Navier--Stokes equations evolving from arbitrarily large $-1$--homogeneous initial data and present numerical evidence for their non-uniqueness.

This paper deals with the interactions of waves governed by a non-linear dispersive Boussinesq type system with the vertical displacement of a cylindrical floating structure in an axisymmetric without swirl situation. The Boussinesq regime…

We show here that the quasilinear Kirchhoff-Pokhozhaev equation $$u_{tt}-\big(a\int_{\mathbb{R}^n} |\nabla u |^2 dx + b \big)^{-2} \Delta u = 0,$$ with $a\neq0$, admits conservation laws of all orders.

We consider Lipschitz solutions to the possibly highly degenerate elliptic equation $ {\rm div} G(\nabla u)=0$ in $B_1\subset\mathbb{R}^2 $, for any continuous strictly monotone vector field $G \colon \mathbb{R}^2 \to \mathbb{R}^2$. We show…

We study the Choquard equation involving mixed local and nonlocal operators \[-\varepsilon^{2}\Delta u+\varepsilon^{2s}(-\Delta)^{s}u+V(x)u=\varepsilon^{\mu-2}\left(\frac{1}{|x|^{\mu}}*F(u)\right)f(u)\quad \text{in }\R^{2},\] where…

We study fractional evolution equations driven by rational-kernel time operators with non-singular memory, including the Atangana-Baleanu-Caputo operator and a generalized W-operator. These operators are characterized by Laplace symbols…

We introduce a new two-parameter fractional time operator with Volterra structure, denoted by the W-operator, defined through a generalized Laplace symbol. The operator preserves the Caputo-type high-frequency behavior while allowing a…

We investigate the Cauchy-Dirichlet problem for linear parabolic equations in divergence form. Under mild assumptions on the source term and the domain, we prove the existence of globally H\"{o}lder continuous solutions. Notably, our…