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The existence and uniqueness of weak solutions to dynamical low-rank evolution problems for parabolic partial differential equations in two spatial dimensions is shown, covering also non-diagonal diffusion in the elliptic part. The proof is…

We prove that if a parabolic Lipschitz (i.e., Lip(1,1/2)) graph domain has the property that its caloric measure is a parabolic $A_\infty$ weight with respect to surface measure (which in turn is equivalent to $L^p$ solvability of the…

We present a family of integral equation-based solvers for the linear or semilinear heat equation in complicated moving (or stationary) geometries. This approach has significant advantages over more standard finite element or finite…

We study the Cauchy problem for the fractional semilinear heat equation with distributional inhomogeneous terms. By introducing the Lorentz--Morrey spaces, we overcome limitations of real interpolation in the classical local Morrey spaces…

The paper is concerned with the Cauchy problem for a semi-linear hyperdissipative heat equation in Besov and Triebel-Lizorkin spaces which is related to the generalized Gauss-Weierstrass semi-group via Duhamel's principle. Using caloric…

The H\"older continuity of the solution to a nonlinear stochastic partial differential equation arising from one dimensional super process is obtained. It is proved that the H\"older exponent in time variable is as close as to 1/4,…

We examine various density results related to the solutions of the non-local heat equation at a specific time slice, focusing on two distinct models: one with homogeneous Dirichlet boundary condition and the other with singular boundary…

A new class of solvable $N$-body problems is identified. They describe $N$ unit-mass point particles whose time-evolution, generally taking place in the complex plane, is characterized by Newtonian equations of motion "of goldfish type"…

We consider the Dirichlet problem for the nonlinear $p(x)$-Laplacian equation. For axially symmetric domains we prove that, under suitable assumptions, there exist Mountain-pass solutions which exhibit partial symmetry. Furthermore, we show…

This paper is twofold. The first part aims to study the long-time asymptotic behavior of solutions to the heat equation on Riemannian symmetric spaces $G/K$ of noncompact type and of general rank. We show that any solution to the heat…

Inspired by the Galerkin and particular method, a new approximation approach is recalled in the Cartesian case. In this paper, we are interested specially by constructing this method, when the domain of consideration is a two dimensional…

We consider the inhomogeneous heat equation on the half-space $\mathbb R_{+}^{d}$ with Neumann boundary conditions. We prove a space-time Gevrey regularity of the solution, with a radius of analyticity uniform up to the boundary of the…

We consider the so-called \emph{discrete $p$-Laplacian}, a nonlinear difference operator that acts on functions defined on the nodes of a possibly infinite graph. We study the associated nonlinear Cauchy problem and identify the generator…

In this work, we study the asymptotic behaviour of solutions to the heat equation in exterior domains, i.e., domains which are the complement of a smooth compact set in $\mathbb{R}^N$. Different homogeneous boundary conditions are…

We present a class of thermodynamic systems with constant thermodynamic curvature which, within the context of geometric approaches of thermodynamics, can be interpreted as constant thermodynamic interaction among their components. In…

We consider the semilinear heat equation $$ u_t-\Delta u=|u|^{p-1}u,\ \ (t,x)\in\mathbb{R}^+\times\mathbb{R}^n. $$ The well-known difficulty with this problem is that the potential well method cannot be applied directly, due to the scaling…

The paper examines the existence of pseudo almost periodic solutions to some classes of partial hyperbolic evolution equations. Namely, sufficient conditions for the existence and uniqueness of pseudo almost periodic solutions to those…

The parabolic Anderson problem is the Cauchy problem for the heat equation $\partial_t u(t,z)=\Delta u(t,z)+\xi(z) u(t,z)$ on $(0,\infty)\times {\mathbb Z}^d$ with random potential $(\xi(z) \colon z\in {\mathbb Z}^d)$. We consider…

We study the one-dimensional one-phase Stefan problem for the heat equation with a nonlinear boundary condition. We show that all solutions fall into one of three distinct types: global-in-time solutions with exponential decay,…

In this article we develop an analogue of Aubry Mather theory for time periodic dissipative equation \[ \left\{ \begin{aligned} \dot x&=\partial_p H(x,p,t),\\ \dot p&=-\partial_x H(x,p,t)-f(t)p \end{aligned} \right. \] with $(x,p,t)\in…