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On the evolution of subcritical regions for the Perona-Malik equation
Marina Ghisi, Massimo Gobbino
The Perona-Malik equation is a celebrated example of forward-backward parabolic equation. The forward behavior takes place in the so-called subcritical region, in which the gradien…
An example of global classical solution for the Perona-Malik equation
Marina Ghisi, Massimo Gobbino
We consider the Cauchy problem for the Perona-Malik equation in an open subset of R^{n}, with Neumann boundary conditions. It is well known that in the one-dimensional case this pr…
Mildly degenerate Kirchhoff equations with weak dissipation: global existence and time decay
Marina Ghisi, Massimo Gobbino
We consider the hyperbolic-parabolic singular perturbation problem for a degenerate quasilinear Kirchhoff equation with weak dissipation. This means that the coefficient of the dis…
Hyperbolic--parabolic singular perturbation for nondegenerate Kirchhoff equations with critical weak dissipation
Marina Ghisi, Massimo Gobbino
We consider the hyperbolic-parabolic singular perturbation problem for a nondegenerate quasilinear equation of Kirchhoff type with weak dissipation. This means that the dissipative…
Spectral gap global solutions for degenerate Kirchhoff equations
Marina Ghisi, Massimo Gobbino
We consider the second order Cauchy problem where is a continuous function, and is a self-ad…
A uniqueness result for Kirchhoff equations with non-Lipschitz nonlinear term
Marina Ghisi, Massimo Gobbino
We consider the second order Cauchy problem $$u''+\m{u}Au=0, u(0)=u_{0}, u'(0)=u_{1},$$ where is a continuous function, and is a self-adjoint nonn…