A celebrated conjecture due to De Giorgi states that any bounded solution of the equation ∆u + (1 -u 2 )u = 0 in R N with ∂y N u > 0 must be such that its level sets {u = λ} are all hyperplanes, at least for dimension N ≤ 8.A counterexample for N ≥ 9 has long been believed to exist.Starting from a minimal graph Γ which is not a hyperplane, found by Bombieri, De Giorgi and Giusti in R N , N ≥ 9, we prove that for any small α > 0 there is a bounded solution uα(y) with ∂y N uα > 0, which resembles tanhä , where t = t(y) denotes a choice of signed distance to the blown-up minimal graph Γα := α -1 Γ.This solution is a counterexample to De Giorgi's conjecture for N ≥ 9.
Starting from a bound state (positive or sign-changing) solution to $$ -\Delta \omega_m =|\omega_m|^{p-1} \omega_m -\omega_m \ \ \mbox{in}\ \R^n, \ \omega_m \in H^2 (\R^n)$$ and solutions to the Helmholtz equation $$ \Delta u_0 + \lambda u_0=0 \ \ \mbox{in} \ \R^n, \ \lambda>0, $$ we build new Dancer's type entire solutions to the nonlinear scalar equation $$ -\Delta u =|u|^{p-1} u-u \ \ \mbox{in} \ \R^{m+n}. $$
The Γ-convergence theory shows that under certain conditions the diblock copolymer equation has spot and ring solutions. We determine the asymptotic properties of the critical eigenvalues of these solutions in order to understand their stability. In two dimensions a threshold exists for the stability of the spot solution. It is stable if the sample size is small and unstable if the sample size is large. The stability of the ring solutions is reduced to a family of finite dimensional eigenvalue problems. In one study no two-interface ring solutions are found by the Γ-convergence method if the sample is small. A stable two-interface ring solution exists if the sample size is increased. It becomes unstable if the sample size is increased further.
A semilinear elliptic equation on a bounded domain in R2 with large exponent in the nonlinear term is studied in this paper.We investigate positive solutions obtained by the variational method.It turns put that the constrained minimizing problem possesses nice asymptotic behavior as the nonlinear exponent, serving as a parameter, gets large.We shall prove that cp , the minimum of energy functional with the nonlinear exponent equal to p , is like (&Ke)lf2p~^2 as p tends to infinity.Using this result, we shall prove that the variational solutions remain bounded uniformly in p .As p tends to infinity, the solutions develop one or two peaks.Precisely the solutions approach zero except at one or two points where they stay away from zero and bounded from above.Then we consider the problem on a special class of domains.It turns out that the solutions then develop only one peak.For these domains, the solutions enlarged by a suitable quantity behave like a Green's function of -A.In this case we shall also prove that the peaks must appear at a critical point of the Robin function of the domain.ÍAu + up = 0 inilci?",n>3, \"lan = 0.They showed that when p approaches (n + 2)/(n -2), the well-known critical exponent, the positive solutions obtained by the variational method will blow up at some point xo which is a critical point of function R where R{x) = g(x, x), and g(x, y) is the regular part of the Green's function of -A.
A linear stability analysis of localized spike solutions to the singularly perturbed two-component Gierer--Meinhardt (GM) reaction-diffusion (RD) system with a fixed time delay $T$ in the nonlinear reaction kinetics is performed. Our analysis of this model is motivated by the computational study of Lee, Gaffney, and Monk [Bull. Math. Bio., 72 (2010), pp. 2139--2160] on the effect of gene expression time delays on spatial patterning for both the GM model and some related RD models. It is shown that the linear stability properties of such localized spike solutions are characterized by the discrete spectra of certain nonlocal eigenvalue problems (NLEP). Phase diagrams consisting of regions in parameter space where the steady-state spike solution is linearly stable are determined for various limiting forms of the GM model in both 1-dimensional and 2-dimensional domains. On the boundary of the region of stability, the spike solution is found to undergo a Hopf bifurcation. For a special range of exponents in the nonlinearities for the 1-dimensional GM model, and assuming that the time delay only occurs in the inhibitor kinetics, this Hopf bifurcation boundary is readily determined analytically. For this special range of exponents, the challenging problem of locating the discrete spectrum of the NLEP is reduced to the much simpler problem of locating the roots to a simple transcendental equation in the eigenvalue parameter. By using a hybrid analytical-numerical method, based on a parametrization of the NLEP, it is shown that qualitatively similar phase diagrams occur for general GM exponent sets and for the more biologically relevant case where the time delay occurs in both the activator and inhibitor kinetics. Overall, our results show that there is a critical value $T_{\star}$ of the delay for which the spike solution is unconditionally unstable for $T>T_{*}$, and that the parameter region where linear stability is assured is, in general, rather limited. A comparison of the theory with full numerical results computed from the RD system with delayed reaction kinetics for a particular parameter set suggests that the Hopf bifurcation can be subcritical, leading to a global breakdown of a robust spatial patterning mechanism.
One of the most important findings in the study of chemotactic process is self-organized cellular aggregation, and a high volume of results are devoted to the analysis of a concentration of single species. Whereas, the multi-species case is not understood as well as the single species one. In this paper, we consider two-species chemotaxis systems with logistic source in a bounded domain $\Omega\subset \mathbb R^2.$ Under the large chemo-attractive coefficients and one certain type of chemical production coefficient matrices, we employ the inner-outer gluing approach to construct multi-spots steady states, in which the profiles of cellular densities have strong connections with the entire solutions to Liouville systems and their locations are determined in terms of reduced-wave Green's functions. In particular, some numerical simulations and formal analysis are performed to support our rigorous studies.
A classical problem in fluid dynamics concerns the interaction of multiple vortex rings sharing a common axis of symmetry in an incompressible, inviscid $3$-dimensional fluid. Helmholtz (1858) observed that a pair of similar thin, coaxial vortex rings may pass through each other repeatedly due to the induced flow of the rings acting on each other. This celebrated configuration, known as leapfrogging, has not yet been rigorously established. We provide a mathematical justification for this phenomenon by constructing a smooth solution of the 3d Euler equations exhibiting this motion pattern.
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