Solvability of boundary value problems for strongly degenerate parabolic equations with discontinuous coefficients

In this paper, we prove the existence of Radon measure-valued solutions for nonlinear degenerate parabolic equations with nonnegative bounded Radon measure data. Moreover, we show the uniqueness of the measure-valued solutions when the Radon measure as a forcing term is diffuse with respect to the parabolic capacity and the Radon measure as a initial value is diffuse with respect to the Newtonian capacity. We also deduce that the concentrated part of the solution with respect to the Newtonian capacity depends on time.

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The cost of controlling strongly degenerate parabolic equations

ESAIM Control Optimisation and Calculus of Variations ◽

10.1051/cocv/2018007 ◽

2020 ◽

Vol 26 ◽

pp. 2

Author(s):

P. Cannarsa ◽

P. Martinez ◽

J. Vancostenoble

Keyword(s):

Parabolic Equations ◽

Complex Analysis ◽

Degenerate Parabolic Equations ◽

Control Cost ◽

Locally Distributed Control ◽

Degenerate Parabolic ◽

The Moment ◽

The Cost ◽

Strongly Degenerate ◽

Degeneracy Parameter

We consider the typical one-dimensional strongly degenerate parabolic operator Pu = ut − (xαux)x with 0 < x < ℓ and α ∈ (0, 2), controlled either by a boundary control acting at x = ℓ, or by a locally distributed control. Our main goal is to study the dependence of the so-called controllability cost needed to drive an initial condition to rest with respect to the degeneracy parameter α. We prove that the control cost blows up with an explicit exponential rate, as eC/((2−α)2T), when α → 2− and/or T → 0+. Our analysis builds on earlier results and methods (based on functional analysis and complex analysis techniques) developed by several authors such as Fattorini-Russel, Seidman, Güichal, Tenenbaum-Tucsnak and Lissy for the classical heat equation. In particular, we use the moment method and related constructions of suitable biorthogonal families, as well as new fine properties of the Bessel functions Jν of large order ν (obtained by ordinary differential equations techniques).

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Remarks on global solutions to the initial-boundary value problem for quasilinear degenerate parabolic equations with a nonlinear source term

Opuscula Mathematica ◽

10.7494/opmath.2019.39.3.395 ◽

2019 ◽

Vol 39 (3) ◽

pp. 395-414

Author(s):

Mitsuhiro Nakao

Keyword(s):

Boundary Value Problem ◽

Parabolic Equations ◽

Boundary Value ◽

Global Solutions ◽

Positive Function ◽

Initial Boundary ◽

Initial Boundary Value Problem ◽

Degenerate Parabolic Equations ◽

Degenerate Parabolic ◽

Initial Boundary Value

We give an existence theorem of global solution to the initial-boundary value problem for \(u_{t}-\operatorname{div}\{\sigma(|\nabla u|^2)\nabla u\}=f(u)\) under some smallness conditions on the initial data, where \(\sigma (v^2)\) is a positive function of \(v^2\ne 0\) admitting the degeneracy property \(\sigma(0)=0\). We are interested in the case where \(\sigma(v^2)\) has no exponent \(m \geq 0\) such that \(\sigma(v^2) \geq k_0|v|^m , k_0 \gt 0\). A typical example is \(\sigma(v^2)=\operatorname{log}(1+v^2)\). \(f(u)\) is a function like \(f=|u|^{\alpha} u\). A decay estimate for \(\|\nabla u(t)\|_{\infty}\) is also given.

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