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Asymptotic axial symmetry of solutions of parabolic equations in bounded radial domains. (English) Zbl 1259.35012

In the celebrated paper by B. Gidas et al. [Commun. Math. Phys. 68, 209–243 (1979; Zbl 0425.35020)], the following result was proved: Let \(u \in C^2(\overline \Omega)\) be a positive solution of \(\Delta u + f(u) = 0\) in the ball \(B = \{\, |x| < R \,\} \subset \mathbb R^N\), vanishing on \(\partial B\). Here \(f\) can be any function of class \(C^1\). Then \(u\) is radially symmetric and \(\partial u / \partial r < 0\) for \(r = |x| \in (0,R)\). In the paper under review, the authors drop the assumption that \(u\) is positive. Instead, they assume that \(u\) satisfies a reflection inequality with respect to a hyperplane containing the origin. The precise assumption is the following: (U1) There exists a unit vector \(e\) such that \(u(x) \geq u(\sigma_e(x))\) for all \(x\) in the half-ball \(B(e) = B \cap \{\, x \cdot e > 0 \,\}\), and \(u(x) \not \equiv u(\sigma_e(x))\). Here \(\sigma_e(x) = x - 2 \, (x \cdot e) \, e\) denotes reflection with respect to the hyperplane \(x \cdot e = 0\). The conclusion is that \(u\) is foliated Schwarz symmetric with respect to some unit vector \(p\), i.e., \(u\) is axially symmetric with respect to the axis \(\mathbb R \, p\) and nonincreasing in the angle \(\theta = \arccos(\frac{x}{|x|} \cdot p)\). The name foliated Schwarz symmetry was introduced by D. Smets and M. Willem [Calc. Var. Partial Differ. Equ. 18, No. 1, 57–75 (2003; Zbl 1274.35026)]. Several further results are obtained. For instance, the case when the domain of the problem is an annulus is considered, as well as the case when \(f = f(|x|,u)\). The results follow as a corollary to the authors’ main theorem, dealing with the reaction-diffusion problem \[ \begin{cases} u_t = \Delta u + f(t, \, |x|, \, u), &(x,t) \in B \times (0, +\infty),\\ u(x,t) = 0, &(x,t) \in \partial B \times (0, +\infty),\\ u(x,0) = u_0(x), &x \in B. \end{cases} \] Concerning this problem, the authors investigate the asymptotic behavior of any classical solution \(u\), assuming that \(u\) exists for all \(t \in [0,+\infty)\). More precisely, for every fixed \(t\) the function \(x \mapsto u(x,t)\) is considered. The norms of such functions in \(L^\infty(B)\) are supposed to be uniformly bounded. If there exists a sequence \(t_n \to +\infty\) such that the corresponding functions uniformly converge to some function \(z(x)\) in \(C^0(\overline B)\), then \(z\) is said to belong to the omega limit set \(\omega(u)\). The authors prove that all functions \(z \in \omega(u)\) are foliated Schwarz symmetric, assuming some regularity of \(f\) and also assuming that the datum \(u_0(x)\) satisfies (U1). The symmetry axis of \(z\) is not fixed a priori by that assumption. The proof is based on the rotating plane method, which is a variant of the moving plane method and was used in the elliptic setting by F. Pacella and T. Weth [Proc. Am. Math. Soc. 135, No. 6, 1753–1762 (2007; Zbl 1190.35096)]. The paper is inspired by P. Poláčik [Arch. Ration. Mech. Anal. 183, No. 1, 59–91 (2007; Zbl 1171.35063)], who studied asymptotic symmetry of positive solutions to fully nonlinear parabolic equations.

MSC:

35B06 Symmetries, invariants, etc. in context of PDEs
35B40 Asymptotic behavior of solutions to PDEs
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
35K20 Initial-boundary value problems for second-order parabolic equations
35K58 Semilinear parabolic equations
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References:

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