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This paper is concerned with the uniqueness, existence, partial smoothing effect, comparison principle and long-time behavior of solutions to the initial-boundary value problem for a unidirectional diffusion equation. The unidirectional evolution often appears in Damage Mechanics due to the strong irreversibility of crack propagation or damage evolution. The existence of solutions is proved in an -framework by employing a backward Euler scheme and by introducing a new method of a priori estimates based on a reduction of discretized equations to variational inequalities of obstacle type and by developing a regularity theory for such obstacle problems. The novel discretization argument will be also applied to prove the comparison principle as well as to investigate the long-time behavior of solutions. 相似文献
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Every solution u = u(x, t) of the Cauchy–Dirichlet problem for the fast diffusion equation, ? t (|u| m-2 u) = Δu in Ω × (0, ∞) with a smooth bounded domain Ω of ${\mathbb{R}^N}$ and 2 < m < 2* : = 2N/(N ? 2)+, vanishes in finite time at a power rate. This paper is concerned with asymptotic profiles of sign-changing solutions and a stability analysis of the profiles. Our method of proof relies on a detailed analysis of a dynamical system on some surface in the usual energy space as well as energy method and variational method. 相似文献
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Non-acoustical and acoustical properties of reticulated and partially reticulated polyurethane foams
A series of measurements on four polyurethane foam samples with pore membranes and a polyurethane foam sample without pore membranes have been made. Tortuosity has been deduced using the ultrasonic slope method. It has been found that the deduced value of tortuosity depends on the measurement temperature and for two of the polyurethane foam samples with many pore membranes physically meaningful values of tortuosity cannot be obtained at a temperature of around 25 °C. However more realistic values of tortuosity have been obtained by from measurements at or around the glass transition temperature of polyurethane foam (i.e. at ?20 °C) when using the ultrasonic slope method.Flow resistivity, Young’s moduli and loss factors have been measured also.Vibration of the pore membranes has been observed to influence the effective density and characteristic impedance derived from the surface impedance measured in an impedance tube. This paper discusses relationships between membrane vibration and the slow and fast compressional waves. The relative merits of predictions based on rigid-porous models and the Biot–Johnson–Champoux–Allard model are discussed also. 相似文献
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Goro Akagi 《Journal of Differential Equations》2011,251(7):1790-1812
We discuss the existence of periodic solution for the doubly nonlinear evolution equation A(u′(t))+∂?(u(t))∋f(t) governed by a maximal monotone operator A and a subdifferential operator ∂? in a Hilbert space H. As the corresponding Cauchy problem cannot be expected to be uniquely solvable, the standard approach based on the Poincaré map may genuinely fail. In order to overcome this difficulty, we firstly address some approximate problems relying on a specific approximate periodicity condition. Then, periodic solutions for the original problem are obtained by establishing energy estimates and by performing a limiting procedure. As a by-product, a structural stability analysis is presented for the periodic problem and an application to nonlinear PDEs is provided. 相似文献