相关论文: A comparison principle for higher order nonlinear …
In this note we show a comparison principle for nonlinear heat Rockland operators on graded groups. We give a simple proof for it using purely algebraic relations. As an application of the established comparison principle we prove the…
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…
On graded Lie groups, we develop a mechanism that transfers the uniformity of maximal hypoellipcity from the frozen coefficients principal part of a differential operator to the full operator. Our approach brings the century-old…
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…
This paper discusses the existence of gradient estimates for second order hypoelliptic heat kernels on manifolds. It is now standard that such inequalities, in the elliptic case, are equivalent to a lower bound on the Ricci tensor of the…
We prove the comparison principle for viscosity sub/super-solutions of degenerate subelliptic equations in non-divergence form that include the sub-elliptic infinity Laplacian and the normalized p-Laplacian. The equations are defined by a…
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 heat and wave type equations on a separable Hilbert space $\mathcal{H}$ by considering non-local operators in time with any positive densely defined linear operator with discrete spectrum. We show the explicit representation of the…
We propose analogues of Green's and Picone's identities for the $p$-sub-Laplacian on stratified Lie groups. In particular, these imply a generalised D\'iaz-Sa\'a inequality for the $p$-sub-Laplacian on stratified Lie groups. Using these we…
We study inequalities related to the heat kernel for the hypoelliptic sublaplacian on an H-type Lie group. Specifically, we obtain precise pointwise upper and lower bounds on the heat kernel function itself. We then apply these bounds to…
This note is devoted to several results about frequency localized functions and associated Bernstein inequalities for higher order operators. In particular, we construct some counterexamples for the frequency-localized Bernstein…
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…
We give sufficient conditions on the initial data so that a semilinear wave inequality blows-up in finite time. Our method is based on the study of an associated second order differential inequality. The same method is applied to some…
In this paper, we deal with existence, uniqueness and exact rate of boundary behavior of blow-up solutions for a class of logistic type quasilinear problem in a smooth bounded domain involving the $p$-Laplacian operator, where the…
This paper is concerned with finite blow-up solutions of the heat equation with nonlinear boundary conditions. It is known that a rate of blow-up solutions is the same as the self-similar rate for a Sobolev subcritical case. A goal of this…
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 study the blow-up problem of one-dimensional nonlinear heat equations. Our result shows that for a certain class of initial conditions, the solutions blow up in finite time and we characterize the asymptotic dynamics of these solutions.…
In this paper we study strongly coupled elliptic systems in non-variational form involving fractional Laplace operators. We prove Liouville type theorems and, by mean of the blow-up method, we establish a priori bounds of positive solutions…
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…
We establish sharp higher-order heat estimates with complete bound on the noncommutative tori \(\mathbb{T}_{\theta}^{n}\) and show the optimality in the small-time order. As an application in polynomial semilinear heat equations on…