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2001 Discontinuous limit semigroups for the singular Zhang equation and its hydrodynamic version
Victor A. Galaktionov, Robert Kersner
Adv. Differential Equations 6(12): 1493-1516 (2001).


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.


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Victor A. Galaktionov. Robert Kersner. "Discontinuous limit semigroups for the singular Zhang equation and its hydrodynamic version." Adv. Differential Equations 6 (12) 1493 - 1516, 2001.


Published: 2001
First available in Project Euclid: 2 January 2013

zbMATH: 1009.35046
MathSciNet: MR1858430

Primary: 35K55
Secondary: 35K65

Rights: Copyright © 2001 Khayyam Publishing, Inc.


Vol.6 • No. 12 • 2001
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