Resumen:
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The fully nonlinear parabolic problem (P_{\text{}) u t =min{?,?u} for ?×R + , u=0 for ??×R + , u(x,0)=u 0 (x) for ? , occurs in some cases of Bellman's equation of dynamic programming.
The author studies questions of asymptotic behavior of strong solutions of (P_{\text{}). He proves that u(?,t) converges as t?? , to an equilibrium solution, strongly in H 1 0 (?). The correct equilibrium solution is individuated when some conditions are met by either u 0 (for example ??u 0 ?0 ) or ? (for example ??0 , ???0 ). Instrumental to the above treatment is the study of the problem (P_{\text{}) v t ???(x,v)=0 for ?×R + , ?(x,v)=0 for ??×R + , v(x,0)=v 0 for ? , where ?(x,r)=?min{?,?r} (x??;r?R). Problem (P_{\text{}) is shown to be well posed in L 1 (?). The difficulty here is represented by the fact that ? given above does not meet the standard assumptions that insure that ???(?) is m -accretive in L 1 (?).
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