Related papers: A comparison principle for higher order nonlinear …

Motivated by the problems of analytic hypoellipticity, we show that a special family of compact non self-adjoint operators has a non-zero eigenvalue. We recover old results by Christ,Hanges, Himonas, Pham-The-Lai and Robert proved by using…

Liouville theorems for scaling invariant nonlinear parabolic equations and systems (saying that the equation or system does not possess positive entire solutions) guarantee optimal universal estimates of solutions of related initial and…

Let $m\in\mathbb N$, $P(D):=\sum_{|\alpha|=2m}(-1)^m a_\alpha D^\alpha$ be a $2m$-order homogeneous elliptic operator with real constant coefficients on $\mathbb{R}^n$, and $V$ a measurable function on $\mathbb{R}^n$. In this article, the…

In this paper we apply the differential inequality technique of Payne {\it et. al} \cite{Payne&SchaeferRobin08} to show that a reaction-diffusion system admits blow-up solutions, and to determine an upper bound for the blow-up time. For a…

In this paper, we consider the following indefinite fully fractional heat equation involving the master operator . Under certain assumptions of the indefinite nonlinearity and its weight, we prove that there is no positive bounded solution,…

In this note, we prove a blow-up result for a semilinear generalized Tricomi equation with nonlinear term of derivative type, i.e., for the equation $\mathscr{T}_{\!\!\ell} u = |\partial_t u|^p$, where $ \mathscr{T}_{\!\!\ell} =…

The purpose of this article is to establish upper and lower estimates for the integral kernel of the semigroup exp(-tP) associated to a classical, strongly elliptic pseudodifferential operator P of positive order on a closed manifold. The…

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…

For differential inequalities of the form $$ \sum_{|\alpha| = m} (- 1)^m \partial^\alpha a_\alpha (x, u) \ge b (x) |u|^\lambda \quad \mbox{in } {\mathbb R}^n, \: n \ge 1, $$ where $a_\alpha$ and $b$ are some functions, we obtain conditions…

The group classification of a class of semilinear reaction-diffusion equations with exponential nonlinearity is carried out using the technique of mapping between classes, which was recently proposed in [O.O. Vaneeva, R.O. Popovych and C.…

Motivated by recent works due to Yu--Zhao [J. Geom. Anal. 2020] and Weber--Zacher [arXiv:2012.12974], we study Li--Yau inequalities for the heat equation corresponding to the Dunkl Laplacian, which is a non-local operator parameterized by…

Let $\Omega \ne \emptyset$ be an unbounded open subset of ${\mathbb R}^n$, $n \ge 2$. We obtain blow-up conditions for non-negative solutions of the problem $$ {\mathcal L}_\varphi u \ge F (x, u) \quad \mbox{in } \Omega, \quad \left. u…

We characterize the asymptotic behavior near blowup points for positive solutions of the semilinear heat equation \begin{equation*} \partial_t u-\Delta u =f(u), \end{equation*} for nonlinearities which are genuinely non scale invariant,…

We investigate finite-time blow-up of solutions to the Cauchy problem for a semilinear heat equation posed on infinite graphs. Assuming that the initial datum is sufficiently large, we establish a general blow-up criterion valid on…

A differential-difference operator is used to model the heat equation on a finite graph analogue of Poincar\'e's upper half-plane. Finite analogues of the classical theta functions are shown to be solutions to the heat equation in this…

We produce a finite time blow-up solution for nonlinear fractional heat equation ($\partial_t u + (-\Delta)^{\beta/2}u=u^k$) in modulation and Fourier amalgam spaces on the torus $\mathbb T^d$ and the Euclidean space $\mathbb R^d.$ This…

In this note we obtain gradient estimates for the heat semigroup on nonisotropic Heisenberg groups. More precisely, our aim is to get the H.-Q. Li inequality on nonisotropic Heisenberg groups, which is a generalization of the original…

We establish the comparison principle, existence and regularity of viscosity solutions to the following problem concerning the mixed operator: \begin{align} \begin{cases}…

Given $\alpha > -1$, consider the second order differential operator in $(0,\infty)$, $$L_\alpha f \equiv (x^2 \frac{d^2}{dx^2} + (2\alpha+3)x \frac{d}{dx} + x^2 + (\alpha+1)^2)(f), $$ which appears in the theory of Bessel functions. The…

Gradient inequalities of the Hamilton type and the Li-Yau type for positive solutions to the heat equation are established from a probabilistic viewpoint, which simplifies the proofs of some results of Sun [{\it Pacific J. Math.}, 253…