A Fatou theorem for the equation π’_{π‘}=Ξ(π’-1)β
Marianne Korten Β· Proceedings of the American Mathematical Society Β· 1999
In one space dimension and for a given function u I ( x ) β C 0 β u_I (x) \in C_0 ^ {\infty } (say such that u I ( x ) > 1 u_I (x) > 1 in some interval), the equation u t = Ξ ( u β 1 ) + u_t = \Delta (u-1)_+ can be thought of as describing the energy per unit volume in a Stefan-type problem where the latent heat of the phase change is given by 1 β u I ( x ) 1-u_I (x) . Given a solution 0 β€ u β L l o c 1 ( R n Γ ( 0 , T ) ) 0 \leq u \in L^1 _{\mathrm {loc}} (\mathbb {R} ^n \times (0,T)) to this equation, we prove that for a.e. x 0 β R n x_0 \in \mathbb {R} ^n , there exists