Pseudo-parabolic regularization of forward-backward parabolic equations: Power-type nonlinearities

2016 ◽  
Vol 2016 (712) ◽  
Author(s):  
Michiel Bertsch ◽  
Flavia Smarrazzo ◽  
Alberto Tesei
Keyword(s):  
Parabolic Equations ◽  
Existence And Uniqueness ◽  
Quasilinear Parabolic Equation ◽  
Singular Part ◽  
Qualitative Properties ◽  
Power Type ◽  
Radon Measures ◽  
Entropy Inequalities ◽  
Backward Parabolic Equations ◽  
Quasilinear Parabolic

AbstractWe study a quasilinear parabolic equation of forward-backward type, under assumptions on the nonlinearity which hold for a wide class of mathematical models, using a pseudo-parabolic regularization of power type. We prove existence and uniqueness of positive solutions of the regularized problem in a space of Radon measures. It is shown that these solutions satisfy suitable entropy inequalities. We also study their qualitative properties, in particular proving that the singular part of the solution with respect to the Lebesgue measure is constant in time.

Behavior of blow-up solutions for quasilinear parabolic equations

2020 ◽  
Vol 17 (2) ◽  
pp. 278-295
Author(s):  
Yevgeniia Yevgenieva
Keyword(s):  
Parabolic Equation ◽  
Parabolic Equations ◽  
Blow Up ◽  
Quasilinear Parabolic Equation ◽  
Boundary Function ◽  
Upper Bounds ◽  
Energy Estimates ◽  
Quasilinear Parabolic Equations ◽  
Multidimensional Domain ◽  
Quasilinear Parabolic

We study the quasilinear parabolic equation $(|u|^{q-1}u)_t-\Delta_p\,u=0$ in a multidimensional domain $(0,T)\times\Omega$ under the condition $u(t,x)=f(t,x)$ on $(0,T)\times\partial\Omega$, where the boundary function $f$ blows-up at a finite time $T$, i.e., $f(t,x)\rightarrow\infty$ as $t\rightarrow T$. For $p\geqslant q>0$ and the boundary function $f$ with power-like behavior, the upper bounds of weak solutions of the problem are obtained. The behavior of solutions at the transition from the case where $p>q$ to $p=q$ is investigated. A general approach within the method of energy estimates to such problems is described.

Blow-up of solutions of a quasilinear parabolic equation

2012 ◽  
Vol 142 (2) ◽  
pp. 425-448
Author(s):  
Ryuichi Suzuki ◽  
Noriaki Umeda
Keyword(s):  
Cauchy Problem ◽  
Parabolic Equation ◽  
Parabolic Equations ◽  
Blow Up ◽  
Quasilinear Parabolic Equation ◽  
Positive Function ◽  
Quasilinear Parabolic Equations ◽  
The Cauchy Problem ◽  
Image Position ◽  
Quasilinear Parabolic

We consider non-negative solutions of the Cauchy problem for quasilinear parabolic equations ut = Δum + f(u), where m > 1 and f(ξ) is a positive function in ξ > 0 satisfying f(0) = 0 and a blow-up conditionWe show that if ξm+2/N /(−log ξ)β = O(f(ξ)) as ξ ↓ 0 for some 0 < β < 2/(mN + 2), one of the following holds: (i) all non-trivial solutions blow up in finite time; (ii) every non-trivial solution with an initial datum u0 having compact support exists globally in time and grows up to ∞ as t → ∞: limtt→∞ inf|x|<Ru(x, t) = ∞ for any R > 0. Moreover, we give a condition on f such that (i) holds, and show the existence of f such that (ii) holds.

scholarly journals Perturbative Cauchy theory for a flux-incompressible Maxwell-Stefan system

2022 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Andrea Bondesan ◽  
Marc Briant
Keyword(s):  
Equilibrium State ◽  
Existence And Uniqueness ◽  
Quasilinear Parabolic Equation ◽  
Strong Solutions ◽  
Cross Diffusion ◽  
Anisotropic Norm ◽  
Exponential Trend ◽  
Perturbative Regime ◽  
Global Existence And Uniqueness ◽  
Quasilinear Parabolic

<p style='text-indent:20px;'>Recently, the authors proved [<xref ref-type="bibr" rid="b2">2</xref>] that the Maxwell-Stefan system with an incompressibility-like condition on the total flux can be rigorously derived from the multi-species Boltzmann equation. Similar cross-diffusion models have been widely investigated, but the particular case of a perturbative incompressible setting around a non constant equilibrium state of the mixture (needed in [<xref ref-type="bibr" rid="b2">2</xref>]) seems absent of the literature. We thus establish a quantitative perturbative Cauchy theory in Sobolev spaces for it. More precisely, by reducing the analysis of the Maxwell-Stefan system to the study of a quasilinear parabolic equation on the sole concentrations and with the use of a suitable anisotropic norm, we prove global existence and uniqueness of strong solutions and their exponential trend to equilibrium in a perturbative regime around any macroscopic equilibrium state of the mixture. As a by-product, we show that the equimolar diffusion condition naturally appears from this perturbative incompressible setting.</p>

scholarly journals Existence of extremal periodic solutions for quasilinear parabolic equations

1997 ◽  
Vol 2 (3-4) ◽  
pp. 257-270 ◽  
Author(s):  
Siegfried Carl
Keyword(s):  
Parabolic Equations ◽  
Evolution Equations ◽  
Quasilinear Parabolic Equation ◽  
Upper And Lower Solutions ◽  
Nonlinear Evolution Equations ◽  
Dirichlet Boundary Conditions ◽  
Quasilinear Parabolic Equations ◽  
All Solutions ◽  
Truncation Techniques ◽  
Quasilinear Parabolic

In this paper we consider a quasilinear parabolic equation in a bounded domain under periodic Dirichlet boundary conditions. Our main goal is to prove the existence of extremal solutions among all solutions lying in a sector formed by appropriately defined upper and lower solutions. The main tools used in the proof of our result are recently obtained abstract results on nonlinear evolution equations, comparison and truncation techniques and suitably constructed special testfunction.

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