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In this article we consider the fourth-order operators A1u:= (au")" and A2u:= au"" in divergence and non divergence form, where a: [0, 1] → R+ degenerates in an interior point of the interval. Using the semigroup technique, under suitable assumptions on a, we study the generation property of these operators associated to generalized Wentzell boundary conditions. We prove the well posedness of the corresponding parabolic problems. [ABSTRACT FROM AUTHOR]

We consider operators in divergence form, A1u = (au')', and in nondivergence form, A2u = 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 A1 and A2, equipped with general Wentzell boundary conditions, are nonpositive and selfadjoint on spaces of L² type. [ABSTRACT FROM AUTHOR]

We consider a mixed linear parabolic problem with a second-order strongly elliptic operator and with dynamic boundary conditions, in a domain with smooth boundary $$\varOmega $$ . For each $$\beta \in (0, 1)$$ , we determine necessary and sufficient assumptions on the data, so that there exist a unique solution $$u$$ in $$C^{1+\beta /2,2+\beta }((0, T) \times \varOmega )$$ , with $$D_t u_{|(0, T) \times \partial \varOmega }$$ bounded with values in $$C^{1+\beta }(\partial \varOmega )$$ . [ABSTRACT FROM AUTHOR]

Abstract: Of concern are the initial-boundary value problems for nonautonomous semilinear second order evolution equations with generalized Wentzell boundary conditions. We succeed in establishing a global wellposedness theorem (in a classical sense) for these problems via a specifically designed operator theoretic approach. Moreover, we obtain sharp estimations for the evolution of the solution along the characteristic curves, which enable us to derive the uniform exponential decay of the associated energies. The results obtained in this paper are still new even for the autonomous case. [Copyright &y& Elsevier]