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Global Existence and Asymptotic Stability for a Nonlinear Integrodifferential Equation Modeling Heat Flow

机译：非线性积分微分方程模型热流的全局存在性与渐近稳定性

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We study initial-value problems that arise from models for one-dimensional heat flow (with finite wave speeds) in materials with memory. Under assumptions that ensure compatibility of our constitutive relations with the second law of thermodynamics, the resulting integrodifferential equation is hyperbolic near equilibrium. We establish the existence of unique, global (in time) defined, classical solutions to the problems under consideration, provided the data are smooth and sufficiently close to equilibrium. We treat both Dirichlet and Neumann boundary conditions as well as the problem on the entire real line. Local existence is proved using a contraction-mapping argument which involves estimates for linear hyperbolic PDE's with variable coefficients. Global existence is obtained by deriving a priori energy estimates. These estimates are based on inequalities for strongly positive Volterra kernels (including a new inequality that is needed due to the form of the constitutive relations). Furthermore, compatibility with the second law plays an essential role in the proof in order to obtain an existence result under less restrictive assumptions on the data. (KR)

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