# American Institute of Mathematical Sciences

eISSN:
2688-1594

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## Electronic Research Archive

2019 , Volume 27

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2019, 27: 1-6 doi: 10.3934/era.2019006 +[Abstract](1563) +[HTML](671) +[PDF](313.13KB)
Abstract:

A cluster automorphism is a \begin{document}$\mathbb{Z}$\end{document}-algebra automorphism of a cluster algebra \begin{document}$\mathcal A$\end{document} satisfying that it sends a cluster to another and commutes with mutations. Chang and Schiffler conjectured that a cluster automorphism of \begin{document}$\mathcal A$\end{document} is just a \begin{document}$\mathbb{Z}$\end{document}-algebra homomorphism of a cluster algebra sending a cluster to another. The aim of this article is to prove this conjecture.

2019, 27: 7-19 doi: 10.3934/era.2019007 +[Abstract](1331) +[HTML](586) +[PDF](365.4KB)
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.

2019, 27: 20-36 doi: 10.3934/era.2019008 +[Abstract](1046) +[HTML](508) +[PDF](448.4KB)
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.

2019, 27: 37-67 doi: 10.3934/era.2019009 +[Abstract](1054) +[HTML](483) +[PDF](562.79KB)
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.

2019, 27: 69-87 doi: 10.3934/era.2019010 +[Abstract](650) +[HTML](359) +[PDF](394.36KB)
Abstract:

Let \begin{document}$a,b,c,d,e,f\in\mathbb N$\end{document} with \begin{document}$a\geq c\geq e>0$\end{document}, \begin{document}$b\leq a$\end{document} and \begin{document}$b\equiv a\ ({\rm{mod}}\ 2)$\end{document}, \begin{document}$d\leq c$\end{document} and \begin{document}$d\equiv c\ ({\rm{mod}}\ 2)$\end{document}, \begin{document}$f\leq e$\end{document} and \begin{document}$f\equiv e\ ({\rm{mod}}\ 2)$\end{document}. If any nonnegative integer can be written as \begin{document}$x(ax+b)/2+y(cy+d)/2+z(ez+f)/2$\end{document} with \begin{document}$x,y,z\in\mathbb Z$\end{document}, then the ordered tuple \begin{document}$(a,b,c,d,e,f)$\end{document} is said to be universal over \begin{document}$\Bbb Z$\end{document}. Recently, Z.-W. Sun found all candidates for such universal tuples over \begin{document}$\Bbb Z$\end{document}. In this paper, we use the theory of ternary quadratic forms to show that 44 concrete tuples \begin{document}$(a,b,c,d,e,f)$\end{document} in Sun's list of candidates are indeed universal over \begin{document}$\mathbb Z$\end{document}. For example, we prove the universality of \begin{document}$(16,4,2,0,1,1)$\end{document} over \begin{document}$\Bbb Z$\end{document} which is related to the form \begin{document}$x^2+y^2+32z^2$\end{document}.

2019, 27: 89-99 doi: 10.3934/era.2019011 +[Abstract](608) +[HTML](315) +[PDF](302.27KB)
Abstract:

This paper deals with the initial value problem of a predator-prey system with dispersal and delay, which does not admit the classical comparison principle. When the initial value has nonempty compact support, the initial value problem formulates that two species synchronously invade the same habitat in population dynamics. By constructing proper auxiliary equations and functions, we confirm the faster invasion speed of two species, which equals to the minimal wave speed of traveling wave solutions in earlier works.

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