We prove the existence of suitably defined weak Radon measure-valued solutions of the homogeneous Dirichlet initial-boundary value problem for a class of strongly degenerate quasilinear parabolic equations. We also prove that: (i) the concentrated part of the solution with respect to the Newtonian capacity is constant; (ii) the total variation of the singular part of the solution (with respect to the Lebesgue measure) is nonincreasing in time. Conditions under which Radon measure-valued solutions of problem (P) are in fact function-valued (depending both on the initial data and on the strength of degeneracy) are also given. © 2013 Springer-Verlag Berlin Heidelberg.
Radon measure-valued solutions of nonlinear strongly degenerate parabolic equations / Porzio, Maria Michaela; Flavia, Smarrazzo; Tesei, Alberto. - In: CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS. - ISSN 0944-2669. - STAMPA. - 51:1-2(2014), pp. 401-437. [10.1007/s00526-013-0680-y]
Radon measure-valued solutions of nonlinear strongly degenerate parabolic equations
PORZIO, Maria Michaela;TESEI, Alberto
2014
Abstract
We prove the existence of suitably defined weak Radon measure-valued solutions of the homogeneous Dirichlet initial-boundary value problem for a class of strongly degenerate quasilinear parabolic equations. We also prove that: (i) the concentrated part of the solution with respect to the Newtonian capacity is constant; (ii) the total variation of the singular part of the solution (with respect to the Lebesgue measure) is nonincreasing in time. Conditions under which Radon measure-valued solutions of problem (P) are in fact function-valued (depending both on the initial data and on the strength of degeneracy) are also given. © 2013 Springer-Verlag Berlin Heidelberg.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.