## Advances in Differential Equations

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

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

Victor A. Galaktionov and Robert Kersner

#### 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