Abstract
This paper studies 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 the constitutive relations with the second law of thermodynamics, the resulting integrodifierential equation is hyperbolic near equilibrium. The existence of unique, globally (in time) defined, classical solutions to the problems under consideration is established, provided the data are smooth and sufficiently close to equilibrium. Both Dirichlet and Neumann boundary conditions are treated, 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 partial differential equations 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.
Published Version
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