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

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

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