## Electronic Research Archive

2019 , Volume 27

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**Abstract:**

A cluster automorphism is a

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**Abstract:**

The aim of this paper is to study the time decay of the solutions for two models of the one-dimensional phase-lag thermoelasticity with two temperatures. The first one is obtained when the heat flux vector and the inductive temperature are approximated by a second-order and first-order Taylor polynomial, respectively. In this case, the solutions decay in a slow way. The second model that we consider is obtained taking first-order Taylor approximations for the inductive thermal displacement, the inductive temperature and the heat flux. The decay is, therefore, of exponential type.

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**Abstract:**

This paper treats parabolic final value problems generated by coercive Lax–Milgram operators, and well-posedness is proved for this large class. The result is obtained by means of an isomorphism between Hilbert spaces containing the data and solutions. Like for elliptic generators, the data space is the graph normed domain of an unbounded operator that maps final states to the corresponding initial states, and the resulting compatibility condition extends to the coercive context. Lax–Milgram operators in vector distribution spaces is the main framework, but the crucial tool that analytic semigroups always are invertible in the class of closed operators is extended to unbounded semigroups, and this is shown to yield a Duhamel formula for the Cauchy problems in the set-up. The final value heat conduction problem with the homogeneous Neumann boundary condition on a smooth open set is also proved to be well posed in the sense of Hadamard.

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**Abstract:**

This paper deals with the continuation of solutions to the generalized Camassa-Holm equation with higher-order nonlinearity beyond wave breaking. By introducing new variables, we transform the generalized Camassa-Holm equation to a semi-linear system and establish the global solutions to this semi-linear system, and by returning to the original variables, we obtain the existence of global conservative solutions to the original equation. We introduce a set of auxiliary variables tailored to a given conservative solution, which satisfy a suitable semi-linear system, and show that the solution for the semi-linear system is unique. Furthermore, it is obtained that the original equation has a unique global conservative solution. By Thom's transversality lemma, we prove that piecewise smooth solutions with only generic singularities are dense in the whole solution set, which means the generic regularity.

2018 Impact Factor: 0.263

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