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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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

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

First available in Project Euclid: 2 January 2013

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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


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.

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