Stabilization of ill-posed Cauchy problems for parabolic equations

SummaryIn this paper we study the noncharacteristic Cauchy problem, ut−(a(x)ux)x=0, x∈(0, l), t∈.(0, T], u(0, t)=ϕ(t), ux(0,t)=0, 0≦t≦T, assuming only L∞ for a. In the case of weak a priori bounds on u, we derive stability estimates on u of Hölder type in the interior and of logarithmic type at the boundary. Also the continuous dependence on a is considered.SuntoNel presente lavoro consideriamo il problema di Cauchy non ben posto ut= (a(x)ux)x, x∈(0, l), t∈(0, T), u(0, t)=ϕ(t), ux(0, t)=0, 0≦t≦T. Supponiamo che a sia misurabile e limitato inferiormente e superiormente da constanti positive. Introduciamo delle limitazioni a priori su u e dimostriamo la dipendenza continua di u rispetto al dato ϕ sia in (0, l)×(0, T) (di tipo hölderiano) sia per x=l (di tipo logaritmico). Consideriamo, inoltre, la dipendenza continua di u da a.

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