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Sobolev hyperbola for the periodic parabolic Lane-Emden system. (English) Zbl 1427.35101

Summary: Consider the periodic parabolic Lane-Emden type system \(u_t-\Delta u=a(t)v^p, v_t-\Delta v=b(t)u^q\) in a bounded domain \(\Omega\) of \(\mathbb{R}^3\) with periodic coefficients \(a(t)\) and \(b(t)\), subject to homogeneous Dirichlet boundary condition. It is known that for the elliptic system \(-\Delta u=v^p,-\Delta v=u^q\) in the 3-dimensional (3D) space, there exists a Sobolev hyperbola \(\frac{1}{p+1}+\frac{1}{q+1}=\frac{1}{3}\), which characterizes the existence and nonexistence of nontrivial solutions. In this paper, we show that such a Sobolev hyperbola is also critical to the existence of periodic solutions for the periodic Lane-Emden type system. Moreover, different from the elliptic system, there exists another cure \(pq = 1\), which is singular for the existence of periodic solutions.

MSC:

35K40 Second-order parabolic systems
35K58 Semilinear parabolic equations
35A01 Existence problems for PDEs: global existence, local existence, non-existence
35B10 Periodic solutions to PDEs
35B45 A priori estimates in context of PDEs
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