Advances in Differential Equations

Discontinuous limit semigroups for the singular Zhang equation and its hydrodynamic version

Victor A. Galaktionov and Robert Kersner

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Abstract

We study properties of the solutions of two semilinear parabolic equations, the singular Zhang equation and its hydrodynamic version, $$ u_t=u_{xx}+\ln|u_x|\quad {\rm and} \,\,\,v_t = v_{xx} + {v_x}/v , $$ which exhibit strong singularities on the sets $\{u_x=0\}$ and $\{v = 0\}$ respectively. Using the concept of maximal solutions constructed by a regular monotone approximation, we prove existence and boundedness of such solutions. We show that for some classes of initial data, the solutions become instantly entirely singular: $v(t) \equiv 0$ and $u(t)\equiv -\infty$ for any arbitrarily small $t>0$. In general, we establish that for both equations the solutions cannot satisfy the initial condition in any weak sense; i.e., the corresponding semigroups are not continuous at $t=0$. Such discontinuous semigroups are exhibited by a wide class of singular nonlinear parabolic equations.

Article information

Source
Adv. Differential Equations, Volume 6, Number 12 (2001), 1493-1516.

Dates
First available in Project Euclid: 2 January 2013

Permanent link to this document
https://projecteuclid.org/euclid.ade/1357139956

Mathematical Reviews number (MathSciNet)
MR1858430

Zentralblatt MATH identifier
1009.35046

Subjects
Primary: 35K55: Nonlinear parabolic equations
Secondary: 35K65: Degenerate parabolic equations

Citation

Galaktionov, Victor A.; Kersner, Robert. Discontinuous limit semigroups for the singular Zhang equation and its hydrodynamic version. Adv. Differential Equations 6 (2001), no. 12, 1493--1516. https://projecteuclid.org/euclid.ade/1357139956


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