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Positive solutions to a nonlinear fourth-order partial differential equation. (English) Zbl 1253.35123

Summary: A class of fourth-order partial differential equations with Dirichlet boundary conditions which has important applications in phase transformation theory and in the description of the motion of a very thin layer of viscous incompressible fluid is studied. By transforming the higher order problem into a second-order elliptic system and using a fixed point method, the existence and uniqueness of steady state solutions are obtained. Moreover, by using a semi-discrete method, the existence of strong solutions of the evolution equation is obtained if the initial function is in the neighborhood of a positive steady state solution. Finally, the solution of the evolution equation converges to its steady state solution as the time \(t\to +\infty \).

MSC:

35Q35 PDEs in connection with fluid mechanics
35K25 Higher-order parabolic equations
35B40 Asymptotic behavior of solutions to PDEs
35D35 Strong solutions to PDEs
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