Abstract
We consider operators in divergence form, $A_1u=(au')'$, and in nondivergence form, $A_2u=au''$, provided that the coefficient $a$ vanishes in an interior point of the space domain. Characterizing the domain of the operators, we prove that, under suitable assumptions, the operators $A_1$ and $A_2$, equipped with general Wentzell boundary conditions, are nonpositive and selfadjoint on spaces of $L^2$ type.
Highlights
It is well known that degenerate parabolic equations are widely used as mathematical models in the applied sciences to describe the evolution in time of a given system
In recent years an increasing interest has been devoted to the study of second order differential degenerate operators in divergence or in nondivergence form
To our best knowledge, [16] is the first paper treating the existence of a solution for the Cauchy problem associated to a parabolic equation which degenerates in the interior of the spatial domain in the space L2(0, 1), while in [9] both the degenerate operators A1 and A2 in the space L2(0, 1), with or without weight, were examined
Summary
It is well known that degenerate parabolic equations are widely used as mathematical models in the applied sciences to describe the evolution in time of a given system. For any weakly degenerate a ∈ C[0, 1], let us introduce the following weighted spaces: Ha1(0, 1) := {u ∈ L2(0, 1) For any strongly degenerate a ∈ W 1,∞(0, 1), let us introduce the corresponding weighted spaces
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