Cauchy problem for doubly nonlinear parabolic equations with initial data measures

AbstractIn this article we prove a Harnack inequality for non-negative weak solutions to doubly nonlinear parabolic equations of the form $$\begin{aligned} \partial _t u - {{\,\mathrm{div}\,}}{\mathbf {A}}(x,t,u,Du^m) = {{\,\mathrm{div}\,}}F, \end{aligned}$$ ∂ t u - div A ( x , t , u , D u m ) = div F , where the vector field $${\mathbf {A}}$$ A fulfills p-ellipticity and growth conditions. We treat the slow diffusion case in its full range, i.e. all exponents $$m > 0$$ m > 0 and $$p>1$$ p > 1 with $$m(p-1) > 1$$ m ( p - 1 ) > 1 are included in our considerations.

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Quasilinearization Methods for Nonlinear Parabolic Equations with Functional Dependence

Georgian Mathematical Journal ◽

10.1515/gmj.2002.431 ◽

2002 ◽

Vol 9 (3) ◽

pp. 431-448

Author(s):

A. Bychowska

Keyword(s):

Cauchy Problem ◽

Parabolic Equations ◽

Unknown Function ◽

Functional Dependence ◽

Nonlinear Parabolic Equations ◽

Quasilinearization Method ◽

Convergence Theorems ◽

Nonlinear Parabolic ◽

Quasilinearization Methods ◽

Derivatives Of

Abstract We consider a Cauchy problem for nonlinear parabolic equations with functional dependence. We prove convergence theorems for a general quasilinearization method in two cases: (i) the Hale functional acting only on the unknown function, (ii) including partial derivatives of the unknown function.

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Positivity of solutions to the Cauchy problem for linear and semilinear biharmonic heat equations

Advances in Nonlinear Analysis ◽

10.1515/anona-2020-0138 ◽

2020 ◽

Vol 10 (1) ◽

pp. 353-370 ◽

Cited By ~ 1

Author(s):

Hans-Christoph Grunau ◽

Nobuhito Miyake ◽

Shinya Okabe

Keyword(s):

Cauchy Problem ◽

Initial Data ◽

Fourth Order ◽

Parabolic Equations ◽

Sufficient Conditions ◽

Nonlinear Parabolic Equations ◽

Heat Equations ◽

Cauchy Problems ◽

The Cauchy Problem ◽

Positivity Of Solutions

Abstract This paper is concerned with the positivity of solutions to the Cauchy problem for linear and nonlinear parabolic equations with the biharmonic operator as fourth order elliptic principal part. Generally, Cauchy problems for parabolic equations of fourth order have no positivity preserving property due to the change of sign of the fundamental solution. One has eventual local positivity for positive initial data, but on short time scales, one will in general have also regions of negativity. The first goal of this paper is to find sufficient conditions on initial data which ensure the existence of solutions to the Cauchy problem for the linear biharmonic heat equation which are positive for all times and in the whole space. The second goal is to apply these results to show existence of globally positive solutions to the Cauchy problem for a semilinear biharmonic parabolic equation.

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Local Hölder continuity for some doubly nonlinear parabolic equations in measure spaces

Nonlinear Analysis ◽

10.1016/j.na.2012.11.022 ◽

2013 ◽

Vol 79 ◽

pp. 156-175 ◽

Cited By ~ 4

Author(s):

Eurica Henriques ◽

Rojbin Laleoglu

Keyword(s):

Parabolic Equations ◽

Hölder Continuity ◽

Nonlinear Parabolic Equations ◽

Holder Continuity ◽

Nonlinear Parabolic ◽

Measure Spaces ◽

Doubly Nonlinear ◽

Local Hölder Continuity

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Intrinsic scaling method for doubly nonlinear parabolic equations and its application

Advances in Calculus of Variations ◽

10.1515/acv-2020-0109 ◽

2021 ◽

Vol 0 (0) ◽

Author(s):

Masashi Misawa ◽

Kenta Nakamura

Keyword(s):

Parabolic Equation ◽

Parabolic Equations ◽

Large Data ◽

Nonlinear Parabolic Equations ◽

Sobolev Type ◽

Scaling Method ◽

Nonlinear Parabolic ◽

Doubly Nonlinear ◽

Doubly Nonlinear Equation ◽

Doubly Nonlinear Parabolic Equation

Abstract In this article, we consider a fast diffusive type doubly nonlinear parabolic equation, called 𝑝-Sobolev type flows, and devise a new intrinsic scaling method to transform the prototype doubly nonlinear equation to the 𝑝-Sobolev type flows. As an application, we show the global existence and regularity for the 𝑝-Sobolev type flows with large data.

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