Related papers: Doubling Properties of Caloric Functions
This paper is devoted to studying the local behavior of non-negative weak solutions to the doubly non-linear parabolic equation \begin{equation*} \partial_t u^q - \text{div}\big(|D u|^{p-2}D u\big) = 0 \end{equation*} in a space-time…
In this paper, the versions of trigonometric functions of certain known inequalities for hyperbolic ones are proved, and then corresponding inequalities for means are presented.
We prove explicit doubling inequalities and obtain uniform upper bounds (under $(d-1)$-dimensional Hausdorff measure) of nodal sets of weak solutions for a family of linear elliptic equations with rapidly oscillating periodic coefficients.…
This paper provides a local and global Calder\'on-Zygmund type estimate of a weak solution to the parabolic double-phase system. The proof of local estimate is based on comparison estimates and the scaling invariant property of the…
We introduce a new self-accelerating wave packet solution of the Schrodinger equation in one dimension. We obtain an exact analytical parabolic cylinder wave for the inverted harmonic potential. We show that truncated parabolic cylinder…
In this article we derive a doubling inequality at the boundary for solutions to the Kirchhoff-Love isotropic plate's equation satisfying supported boundary conditions. To this end, we combine the use of a suitable conformal mapping which…
Global and local weighted Gagliardo-Nirenberg inequalities with doubling measures are established. These inequalities are key ingredients for the regularity theory and existence of strong solutions for strongly coupled parabolic and…
Assuming that the Lam\'{e} moduli $\mu$, $\lambda$ are $C^{\tiny{1}}$ and $n\geq2$, we prove quantitative estimates of a weak sense of strong unique continuation for thesolutions of the n-dimensional Lam\'{e} system of the form of three…
In this paper, we introduce and analyse an explicit formulation of fractional powers of the parabolic Lam\'e operator $\mathbb{H}$ and we then study the extension problem associated to such non-local operators. We also study the various…
We find new quantitative estimates on the space-time analyticity of solutions to linear parabolic equations with analytic coefficients near the initial time. We apply the estimates to obtain observability inequalities and…
In this paper, we are exploring some of the properties of the self-similar solutions of the first kind. In particular, we shall discuss the kinematic properties and also check the singularities of these solutions. We discuss these…
We introduce a notion of quasilinear parabolic equations over metric measure spaces. Under sharp structural conditions, we prove that local weak solutions are locally bounded and satisfy the parabolic Harnack inequality. Applications…
This article presents a comprehensive overview and supplement to recent developments in second-order elliptic partial differential equations formulated in double divergence form, along with an exploration of their parabolic counterparts.
We investigate uniqueness, in suitable weighted Lebesgue spaces, of solutions to a class of fractional parabolic and elliptic equations.
We study the local regularity of $p$-caloric functions or more generally of $\phi$-caloric functions. In particular, we study local solutions of non-linear parabolic systems with homogeneous right hand side, where the leading terms has…
In this brief note we show that under a volume non-preserving scaling it is possible to recover the basics for a regularity theory regarding local weak solutions to a parabolic fully anisotropic equation. We characterize self-similar…
In this paper we describe some recent works on quantitative unique continuation for elliptic, parabolic and dispersive equations. The elliptic results are joint work with J.Bourgain, while the remainder of the works discussed are joint…
In this paper, by investigating the monotonicity of a function composed of $% \left( \sinh x\right) /x$ and $\cosh x$ with two parameters in $x$ on $% \left( 0,\infty \right) $, we prove serval theorems related to inequalities for…
Some sharp two-sided Tur\'an type inequalities for parabolic cylinder functions and Tricomi confluent hypergeometric functions are deduced. The proofs are based on integral representations for quotients of parabolic cylinder functions and…
We consider the Cauchy problem for doubly non-linear degenerate parabolic equations on Riemannian manifolds of infinite volume, or in $\R^N$. The equation contains a weight function as a capacitary coefficient which we assume to decay at…