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Nonlinear Degenerate Evolution Equations and Partial Differential Equations of Mixed Type

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  • The Cauchy problem for the evolution equation Mu’(t) + N(t,u(t)) = 0 is studied, where M and N(t,•) are, respectively, possibly degenerate and nonlinear monotone operators from a vector space to its dual. Sufficient conditions for existence and for uniqueness of solutions are obtained by reducing the problem to an equivalent one in which M is the identity but each N(t,•) is multivalued and accretive in a Hilbert space. Applications include weak global solutions of boundary value problems with quasilinear partial differential equations of mixed Sobolev-parabolic-elliptic type, boundary conditions with mixed space-time derivatives, and those of the fourth or fifth type. Similar existence and uniqueness results are given for the semilinear and degenerate wave equation Bu"(t) + F(t, u’(t)) + Au(t) = 0, where each nonlinear F(t,•) is monotone and the nonnegative B and positive A are self-adjoint operators from a reflexive Banach space to its dual.
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  • Showalter, R. E. (1975). Nonlinear degenerate evolution equations and partial differential equations of mixed type. SIAM Journal on Mathematical Analysis, 6(1), 25-42. doi:10.1137/0506004
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  • 6
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  • 1
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  • This work was supported in part by National Science Foundation grant GP-34261.
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