Related papers: $L^p$ estimate of the heat equation on a bounded d…

We consider the linear heat equation on a bounded domain and on an exterior domain. We study estimates of any order derivatives of the solution locally in time in the Lebesgue spaces. We give a proof of the estimates in the end-point cases…

Various sharp pointwise estimates for the gradient of solutions to the heat equation are obtained. The Dirichlet and Neumann conditions are prescribed on the boundary of a half-space. All data belong to the Lebesgue space $L^p$. Derivation…

We discuss the unique solvability of the third initial-boundary value problem for the heat equation in a bounded domain. This problem has uniquely a time-global solution in the anisotropic Sobolev space $W^{2,1}_{p,q}$ for any $1<p<\infty$,…

We consider the parabolic Lam\'{e} system on a bounded domain. We focus on two types of inequalities for higher-order derivatives of solutions. The first is related to an $L^p$-$L^p$ estimate locally in time in the Lebesgue space setting,…

For each $1\leq p<\infty$ a Banach space of integrable Schwartz distributions is defined by taking the distributional derivative of all functions in $L^p({\mathbb R})$. Such distributions can be integrated when multiplied by a function that…

Sharp estimates of solutions of the classical heat equation are proved in $L^p$ norms on the real line.

We study the relativistic heat equation in one space dimension. We prove a local regularity result when the initial datum is locally Lipschitz in its support. We propose a numerical scheme that captures the known features of the solutions…

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 study the linear heat equation on a halfspace with a linear dynamical boundary condition. We are interested in an appropriate choice of the function space of initial functions such that the problem possesses a solution. It was known…

This paper is devoted to the study of gradient estimates for the Dirichlet problem of the heat equation in the exterior domain of a compact set. Our results describe the time decay rates of the derivatives of solutions to the Dirichlet…

A heat equation with uncertain domains is thoroughly investigated. Statistical moments of the solution is approximated by the counterparts of the shape derivative. A rigorous proof for the existence of the shape derivative is presented.…

This paper considerers the problem of computing the value of a solution of the heat equation at a given point inside a bounded domain after the initial time. It is assumed that the initial value of the solution inside the domain (possibly…

We establish a refined $L_p$-estimate ($p\geq 2$) for the stochastic heat equation on angular domains in $\mathbb{R}^2$ with mixed weights based on both, the distance to the boundary and the distance to the vertex. This way we can capture…

We study the $L^{p}$-solutions for the semilinear heat equation with unbounded coefficients and driven by a infinite dimensional fractional Brownian motion with self-similarity parameter $H > 1/2$. Existence and uniqueness of local mild…

We establish the local existence and the uniqueness of solutions of the heat equation with a nonlinear boundary condition for the initial data in uniformly local $L^r$ spaces. Furthermore, we study the sharp lower estimates of the blow-up…

In this work we address some questions concerning the Cauchy problem for a generalized nonlinear heat equations considering as functional framework the variable Lebesgue spaces $L^{p(\cdot)}(\mathbb{R}^n)$. More precisely, by mixing some…

We consider an abstract non-negative self-adjoint operator $H$ on an $L^2$-space. We derive a characterization for the restriction estimate $\| dE_H(\lambda) \|_{L^p \to L^{p'}} \le C \lambda^{\frac{d}{2}(\frac{1}{p} - \frac{1}{p'}) -1}$ in…

We prove the $L^p-L^q$ $(1<p\leqslant 2\leqslant q<+\infty)$ norm estimates for the solutions of heat and wave type equations on a locally compact separable unimodular group $G$ by using an integro-differential operator in time and any…

We study pointwise and $L^p$ gradient estimates of the heat kernel, on manifolds that may have some amount of negative Ricci curvature, provided it is not too negative (in an integral sense) at infinity. We also prove uniform boundedness…

We propose a Hilbert space solution theory for a nonhomogeneous heat equation with delay in the highest order derivatives with nonhomogeneous Dirichlet boundary conditions in a bounded domain. Under rather weak regularity assumptions on the…