We prove two-sided estimates of heat kernels on non-parabolic Riemannian manifolds with ends, assuming that the heat kernel on each end separately satisfies the Li-Yau estimate.

The aim of this note is to give a short review of our recent work (see [5]) with Miguel A. Alejo and Luis Vega, concerning the $L^2$-stability, and asymptotic stability, of the $N$-soliton of the Korteweg-de Vries (KdV) equation.

Optimal control problems for the heat equation with pointwise bilateral control-state constraints are considered. A locally superlinearly convergent numerical solution algorithm is proposed and its mesh independence is established. Further, for the efficient numerical solution reduced space and Schur complement based preconditioners are proposed which take into account the active and inactive set structure of the problem. The paper ends by numerical tests illustrating our theoretical findings and...

Optimal control problems for the heat equation with pointwise bilateral control-state constraints are considered. A locally superlinearly convergent numerical solution algorithm is proposed and its mesh independence is established. Further, for the efficient numerical solution reduced space and Schur complement based preconditioners are proposed which take into account the active and inactive set structure of the problem. The paper ends by numerical tests illustrating our theoretical findings and comparing...

We prove the interior null-controllability of one-dimensional parabolic equations with time independent measurable coefficients.

This article is devoted to the numerical study of a flame ball model, derived by Joulin, which obeys to a singular integro-differential equation. The numerical scheme that we analyze here, is based upon a one step method, and we are interested in its long-time behaviour. We recover the same dynamics as in the continuous case: quenching, or stabilization of the flame, depending on heat losses, and an energy input parameter.

This article is devoted to the numerical study of a flame ball model, derived by Joulin, which obeys to a singular integro-differential equation. The numerical scheme that we analyze here, is based upon a one step method, and we are interested in its long-time behaviour. We recover the same dynamics as in the continuous case: quenching, or stabilization of the flame, depending on heat losses, and an energy input parameter.

This work is concerned with the inverse problem of determining initial value of the Cauchy problem for a nonlinear diffusion process with an additional condition on free boundary. Considering the flow of water through a homogeneous isotropic rigid porous medium, we have such desire: for every given positive constants $K$ and $T_0$, to decide the initial value $u_0$ such that the solution $u(x,t)$ satisfies ${sup}_{x\in H_u\left(T_0\right)}\left|x\right|\ge K$, where $H_u\left(T_0\right)=\{x\in {ℝ}^N:u(x,T_0)>0\}$. In this paper, we first establish a priori estimate $u_t\ge C\left(t\right)u$ and a more precise Poincaré type inequality...

Sufficient conditions are obtained so that a weak subsolution of $(0.1)$, bounded from above on the parabolic boundary of the cylinder $Q$, turns out to be bounded from above in $Q$.

We consider the second order parabolic partial differential equation    ${\sum }_{i,j=1}^n{a}_{ij}(x,t)u_{x_i{x}_j}+{\sum }_{i=1}^n{b}_i(x,t)u_{x_i}+c(x,t)u-u_t=0$. Sufficient conditions are given under which every solution of the above equation must decay or tend to infinity as |x|→ ∞. A sufficient condition is also given under which every solution of a system of the form    $L^{\alpha }\left[u^{\alpha }\right]+{\sum }_{\beta =1}^N{c}^{\alpha \beta }(x,t)u^{\beta }=f^{\alpha }(x,t)$, where    $L^{\alpha }\left[u\right]\equiv {\sum }_{i,j=1}^n{a}_{ij}^{\alpha }(x,t)u_{x_i{x}_j}+{\sum }_{i=1}^n{b}_i^{\alpha }(x,t)u_{x_i}-u_t$, must decay as t → ∞.