Related papers: Pointwise Estimates for Relative Fundamental Solut…

We prove the existence and give estimates of the fundamental solution (the heat kernel) for the equation $\partial_t =\mathcal{L}^{\kappa}$ for non-symmetric non-local operators $$ \mathcal{L}^{\kappa}f(x):= \int_{\mathbb{R}^d}(…

We study the large-time behavior of the continuous-time heat kernel and of solutions to the heat equation on homogeneous trees. First, we derive sharp asymptotic formulas for the heat kernel as $t\to\infty$. Second, using them, we show that…

In this paper, we consider the following semi-linear complex heat equation \begin{eqnarray*} \partial_t u = \Delta u + u^p, u \in \mathbb{C} \end{eqnarray*} in $\mathbb{R}^n,$ with an arbitrary power $p,$ $ p > 1$. In particular, $p$ can be…

Some qualitative properties of radially symmetric solutions to the non-homogeneous heat equation with critical density and weighted source $$ |x|^{-2}\partial_tu=\Delta u+|x|^{\sigma}u^p, \quad (x,t)\in\mathbb{R}^N\times(0,T), $$ are…

In this paper, we investigate the pointwise time analyticity of three differential equations. They are the biharmonic heat equation, the heat equation with potentials and some nonlinear heat equations with power nonlinearity of order $p$.…

In this paper, we consider the following complex-valued semilinear heat equation \begin{eqnarray*} \partial_t u = \Delta u + u^p, u \in \mathbb{C}, \end{eqnarray*} in the whole space $\mathbb{R}^n$, where $ p \in \mathbb{N}, p \geq 2$. We…

In this paper, we investigate pointwise time analyticity of solutions to fractional heat equations in the settings of $\mathbb{R}^d$ and a complete Riemannian manifold $\mathrm{M}$. On one hand, in $\mathbb{R}^d$, we prove that any solution…

We prove qualitatively sharp heat kernel bounds in the setting of Fourier-Bessel expansions when the associated type parameter $\nu$ is half-integer. Moreover, still for half-integer $\nu$, we also obtain sharp estimates of all kernels…

Let $L$ be a non-negative self-adjoint operator acting on $L^2(X)$ where $X$ is a space of homogeneous type with a dimension $n$. In this paper, we study sharp endpoint $L^p$-Sobolev estimates for the solution of the initial value problem…

We study existence and uniqueness of weak solutions to (F) $\partial\_t u+ (-\Delta)^\alphau+h(t, u)=0 $ in $(0,\infty)\times\R^N$,with initial condition $u(0,\cdot)=\nu$ in $\R^N$, where $N\ge2$, the operator $(-\Delta)^\alpha$is the…

The theory of pseudo-differential operators is a powerful tool to deal with differential equations involving differential operators under the square root sign. These type of equations are pivotal elements to treat problems in anomalous…

In studies of superlinear parabolic equations \begin{equation*} u_t=\Delta u+u^p,\quad x\in {\mathbb R}^N,\ t>0, \end{equation*} where $p>1$, backward self-similar solutions play an important role. These are solutions of the form $ u(x,t) =…

Let $d\ge1$ and $0<\alpha<2$. Consider the integro-differential operator \[ \mathcal{L}f(x) =\int_{\mathbb{R}^{d}\backslash\{0\}}\left[f(x+h)-f(x)-\chi_{\alpha}(h)\nabla f(x)\cdot…

In this work, we study the heat equation with Grushin's operator. We present an expression for its heat kernel, prove its decay in $L^p$ spaces, and that it is an approximation of the identity. As a consequence, the heat semigroup…

We prove heat kernel estimates for the $\bar\partial$-Neumann Laplacian acting in spaces of differential forms over noncompact, strongly pseudoconvex complex manifolds with a Lie group symmetry and compact quotient. We also relate our…

We are concerned with the following time-fractional semilinear heat equation in the $N$-dimensional whole space ${\bf R}^N$ with $N \geq 1$. \[ {\rm (P)}_\alpha \qquad \partial_t^\alpha u -\Delta u = u^p,\quad t>0,\,\,\, x\in{\bf R}^N,…

We obtain a Li-Yau-type estimate for nonnegative ancient solutions to the subcritical semilinear heat equation $\frac{\p u}{\p t}=\De u+u^p$ in $\rz^n\times(-\infty,0)$. Then, we combine the Li-Yau type estimate and Melre-Zaag's result to…

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 obtain two-sided estimates for the heat kernel (or the fundamental function) associated with the following fractional Schr\"odinger operator with negative Hardy potential $$\Delta^{\alpha/2} -\lambda |x|^{-\alpha}$$ on $\RR^d$, where…

We study the existence and behaviour of blowing-up solutions to the fully fractional heat equation $$ \mathcal{M} u=u^p,\qquad x\in\mathbb{R}^N,\;0<t<T $$ with $p>0$, where $\mathcal{M}$ is a nonlocal operator given by a space-time kernel…