Related papers: Fujita type results for a parabolic inequality wit…
This paper investigates the critical behavior of global solutions to a parabolic equation with a Hartree-type nonlinearity of the form $$\left\{\begin{array}{ll} u_{t}+(-\Delta)^{\frac{\beta}{2}} u= (\mathcal{K}\ast |u|^{p})|u|^{q},&\qquad…
In this paper we investigate the nonexistence of nonnegative solutions of parabolic inequalities of the form $$\begin{cases} &u_t \pm L_\mathcal A u\geq (K\ast u^p)u^q \quad\mbox{ in } \mathbb R^N \times \mathbb (0,\infty),\, N\geq 1,\\…
In this paper, we consider the Cauchy problem for the degenerate parabolic equations on the Heisenberg groups with power law non-linearities. We obtain Fujita-type critical exponents, which depend on the homogeneous dimension of the…
We are concerned with the existence of global and blow-up solutions for the nonlinear parabolic problem described by the Hardy-H\'enon equation $u_t - \Delta_{\mathbb{H}} u = |\cdot|_{\mathbb{H}}^{\gamma} u^p \mbox{ in } \mathbb{H}^N \times…
We consider the fractional order integral equation with a time nonlocal nonlinearity $$^{c}\mathbf{D}_{0\mid t}^{\beta}\left( u \right) +\left(-\Delta_{\mathbb{H}} \right)^{m} \left( u \right) = \frac{1}{\Gamma(\alpha)}\int_{0}^{t}\left(…
We study the uniqueness and nondegeneracy of positive bubble solutions for the generalized energy-critical Hartree equation on the Heisenberg group $\mathbb{H}^{n}$, \begin{equation}\label{0.1}…
In this paper, we will study the following parabolic problem $u_t - div(\omega(x) \nabla u)= h(t) f(u) + l(t) g(u)$ with non-negative initial conditions pertaining to $C_b(\mathbb{R}^N)$, where the weight $\omega$ is an appropriate function…
This paper examines the critical exponents for the existence of global solutions to the equation \begin{equation*} \begin{array}{ll} \displaystyle u_t-\Delta_{\mathbb{H}}u=\int_0^t(t-s)^{-\gamma}|u(s)|^{p-1}u(s)\,ds,&\qquad…
In the Heisenberg group framework, we study rigidity properties for stable solutions of $(-\Delta_H)^s v = f(v)$ in $H$, $s \in (0,1)$. We obtain a Poincar\'e type inequality in connection with a degenerate elliptic equation in $\R^4_+$;…
We prove an invariant Harnack's inequality for operators in non-divergence form structured on Heisenberg vector fields when the coefficient matrix is uniformly positive definite, continuous, and symplectic. The method consists in…
We study regularity results for the solutions of quasilinear subelliptic $p$-Laplace type equation in Heisenberg groups. We prove somewhat surprising excess decay estimates for the constant coefficient homogeneous equation. Excess decay…
In this paper, we study the semilinear heat equation with a forcing term, driven by the fractional sub-Laplacian (-\Delta_{\mathbbm{H}^N})^s of order $s\in (0,1),$ on the Heisenberg group $\mathbbm{H}^N$. We establish that the Fujita…
In this paper, we show the nonexistence results for the Kirchhoff elliptic, parabolic, and hyperbolic type equations on the Heisenberg groups. Also, the pseudo-parabolic and pseudo-hyperbolic equations of the Kirchhoff-type are under…
The aim of this paper is to study the qualitative behaviour of non-negative entire solutions of certain differential inequalities involving gradient terms on the Heisenberg group. We focus our investigation on the two classes of…
The purpose of this paper is to study the existence of weak solutions for some classes of one-parameter subelliptic gradient-type systems involving a Sobolev-Hardy potential defined on an unbounded domain $\Omega_\psi$ of the Heisenberg…
We study a class of inhomogeneous parabolic equations on the Heisenberg group $\mathbbm{H}^N$ with Hardy-type singular potentials, nonlocal memory terms, and a space-time forcing term: \begin{align} \partial_tu-\Delta_{H}u=\lambda…
We consider vector valued weak solutions $u:\Omega_T\to \mathbb{R}^N$ with $N\in \mathbb{N}$ of degenerate or singular parabolic systems of type \begin{equation*} \partial_t u - \mathrm{div} \, a(z,u,Du) = 0 \qquad\text{in}\qquad \Omega_T=…
In this paper we prove symmetry of nonnegative solutions of the integral equation \[ u (\zeta ) = \int\limits_{{\mathbb H}^n} |\zeta^{-1} \xi|^{-(Q-\alpha)} u(\xi)^{p} d\xi \quad 1< p \leq \frac{Q+\alpha}{Q-\alpha},\quad 0< \alpha <Q \] on…
For a general class of divergence type quasi-linear degenerate parabolic equations with measurable coeffcients and lower order terms from non-linear Kato-type classes, we prove local boundedness and continuity of solutions, and the…
We investigate a non-homogeneous nonlinear heat equation which involves degenerate or singular coefficients belonging to the $A_2$ class of functions. We prove the existence of a Fujita exponent and describe the dichotomy…