Related papers: A comparison principle for nonlinear heat Rockland…
In this paper we present a comparison principle for higher order nonlinear hypoelliptic heat operators on graded Lie groups. Moreover, using the comparison principle we obtain blow-up type results and global in $t$-boundedness of solutions…
In this paper, we prove a form of the strong comparison principle for a class of fully nonlinear subelliptic operators of the form $\nabla^{2}_{H}\psi+L(\cdot,\psi,\nabla_{H}\psi)$ on the Heisenberg group, which include the CR invariant…
The complete group classification of a generalization of the Heath model is carried out by connecting it to the heat equation with nonlinear source. Examples of invariant solutions are given under the terminal and the barrier option…
We prove global comparison results for the $p$-Laplacian on a $p$-parabolic manifold. These involve both real-valued and vector-valued maps with finite $p$-energy.
There was proposed the method of a factorization of PDE. The method is based on reduction of complicated systems to more easy ones (for example, due to dimension decrease). This concept is proposed in general case for the arbitrary PDE…
We present extensions of the comparison and maximum principles available for nonlinear non-local integro-differential operators $P:\mathcal{C}^{2,1}(\Omega \times (0,T])\times L^\infty (\Omega \times (0,T])\to\mathbb{R}$, of the form $P[u]…
We prove a comparison theorem for super- and sub-solutions with non-vanishing gradients to semilinear PDEs provided a nonlinearity $f$ is $L^p$ function with $p > 1$. The proof is based on a strong maximum principle for solutions of…
We introduce and study a new class of higher order differential operators defined on $\mathbb{R}^{n}$, which are built with H\"{o}rmander vector fields, homogeneous w.r.t. a family of dilations (but not left invariant w.r.t. any structure…
We present pointwise gradient bounds for solutions to $p$-Laplacean type non-homogeneous equations employing non-linear Wolff type potentials, and then prove similar bounds, via suitable caloric potentials, for solutions to parabolic…
This paper investigates the initial-boundary value problem for a nonlinear parabolic equation involving the $p$-Laplacian operator, nonlocal source terms, gradient absorption, and various nonlinearities: \[ \frac{\partial u}{\partial t} -…
This paper presents some sufficient conditions for the validity of the comparison principle for the weak solutions of non - cooperative weakly coupled systems of elliptic second-order PDEs.
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…
In this paper we use the heat equation in a group of Heisenberg type $\mathbb{G}$ to provide a unified treatment of the two very different extension problems for the time independent pseudo-differential operators $\mathscr L^s$ and…
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 present pointwise gradient bounds for solutions to $p$-Laplacean type non-homogeneous equations employing non-linear Wolff type potentials, and then prove similar bounds, via suitable caloric potentials, for solutions to parabolic…
Let $L = -{\rm div}( A(x) \cdot \nabla ) + V(x)$ be a second-order uniformly elliptic operator on $\mathbb{ R }^{n}$ $(n\geq 3)$, where $A(x)$ is a real symmetric matrix satisfying standard ellipticity conditions, and $V$ is a nonnegative…
We establish the strong comparison principle and strict positivity of solutions to the following nonlinear stochastic heat equation on $\mathbb{R}^d$ \[ \left(\frac{\partial }{\partial t} -\frac{1}{2}\Delta \right) u(t,x) = \rho(u(t,x))…
On stratified Lie groups we study a semilinear heat equation with the Hardy potential, a power non-linearity and a forcing term which depends only upon the spacial variable. The analysis of an equivalent formulation to the problem and an…
A definition of invariance in Lie's sense for a boundary value problem (BVP) with the basic evolution differential equations is proposed. A problem of group classification at a wide class of BVPs parameterized by arbitrary elements is…
In this paper, a new type of comparison theorem is presented for some initial-boundary value problems of second order nonlinear parabolic systems with nonlinear boundary conditions. This comparison theorem has an advantage over the…