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Register a new userExistence of Kazdan-Warner equation with sign-changing prescribed function
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In this paper, we study the following Kazdan-Warner equation with sign-changing prescribed function $h$ begin{align*} -Delta u=8pileft(frac{he^{u}}{int_{Sigma}he^{u}}-1right) end{align*} on a closed Riemann surface whose area is equal to one. The solutions are the critical points of the functional $J_{8pi}$ which is defined by begin{align*} J_{8pi}(u)=frac{1}{16pi}int_{Sigma}| abla u|^2+int_{Sigma}u-lnleft|int_{Sigma}he^{u}right|,quad uin H^1left(Sigmaright). end{align*} We prove the existence of minimizer of $J_{8pi}$ by assuming begin{equation*} Delta ln h^++8pi-2K>0 end{equation*}at each maximum point of $2ln h^++A$, where $K$ is the Gaussian curvature, $h^+$ is the positive part of $h$ and $A$ is the regular part of the Green function. This generalizes the existence result of Ding, Jost, Li and Wang [Asian J. Math. 1(1997), 230-248] to the sign-changing prescribed function case. We are also interested in the blow-up behavior of a sequence $u_{varepsilon}$ of critical points of $J_{8pi-varepsilon}$ with $int_{Sigma}he^{u_{varepsilon}}=1, limlimits_{varepsilonsearrow 0}J_{8pi-varepsilon}left(u_{varepsilon}right)<infty$ and obtain the following identity during the blow-up process begin{equation*} -varepsilon=frac{16pi}{(8pi-varepsilon)h(p_varepsilon)}left[Delta ln h(p_varepsilon)+8pi-2K(p_varepsilon)right]lambda_{varepsilon}e^{-lambda_{varepsilon}}+Oleft(e^{-lambda_{varepsilon}}right), end{equation*}where $p_varepsilon$ and $lambda_varepsilon$ are the maximum point and maximum value of $u_varepsilon$, respectively. Moreover, $p_{varepsilon}$ converges to the blow-up point which is a critical point of the function $2ln h^{+}+A$.
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We consider an evolution problem associated to the Kazdan-Warner equation on a closed Riemann surface $(Sigma,g)$ begin{align*} -Delta_{g}u=8pileft(frac{he^{u}}{int_{Sigma}he^{u}{rm d}mu_{g}}-frac{1}{int_{Sigma}{rm d}mu_{g}}right) end{align*} where the prescribed function $hgeq0$ and $max_{Sigma}h>0$. We prove the global existence and convergence under additional assumptions such as begin{align*} Delta_{g}ln h(p_0)+8pi-2K(p_0)>0 end{align*} for any maximum point $p_0$ of the sum of $2ln h$ and the regular part of the Green function, where $K$ is the Gaussian curvature of $Sigma$. In particular, this gives a new proof of the existence result by Yang and Zhu [Proc. Amer. Math. Soc. 145 (2017), no. 9, 3953-3959] which generalizes existence result of Ding, Jost, Li and Wang [Asian J. Math. 1 (1997), no. 2, 230-248] to the non-negative prescribed function case.
Let $G=(V,E)$ be a finite connected graph, and let $kappa: Vrightarrow mathbb{R}$ be a function such that $int_Vkappa dmu<0$. We consider the following Kazdan-Warner equation on $G$:[Delta u+kappa-K_lambda e^{2u}=0,] where $K_lambda=K+lambda$ and $K: Vrightarrow mathbb{R}$ is a non-constant function satisfying $max_{xin V}K(x)=0$ and $lambdain mathbb{R}$. By a variational method, we prove that there exists a $lambda^*>0$ such that when $lambdain(-infty,lambda^*]$ the above equation has solutions, and has no solution when $lambdageq lambda^ast$. In particular, it has only one solution if $lambdaleq 0$; at least two distinct solutions if $0<lambda<lambda^*$; at least one solution if $lambda=lambda^ast$. This result complements earlier work of Grigoryan-Lin-Yang cite{GLY16}, and is viewed as a discrete analog of that of Ding-Liu cite{DL95} and Yang-Zhu cite{YZ19} on manifolds.
We study the existence of sign-changing solutions to the nonlinear heat equation $partial _t u = Delta u + |u|^alpha u$ on ${mathbb R}^N $, $Nge 3$, with $frac {2} {N-2} < alpha <alpha _0$, where $alpha _0=frac {4} {N-4+2sqrt{ N-1 } }in (frac {2} {N-2}, frac {4} {N-2})$, which are singular at $x=0$ on an interval of time. In particular, for certain $mu >0$ that can be arbitrarily large, we prove that for any $u_0 in mathrm{L} ^infty _{mathrm{loc}} ({mathbb R}^N setminus { 0 }) $ which is bounded at infinity and equals $mu |x|^{- frac {2} {alpha }}$ in a neighborhood of $0$, there exists a local (in time) solution $u$ of the nonlinear heat equation with initial value $u_0$, which is sign-changing, bounded at infinity and has the singularity $beta |x|^{- frac {2} {alpha }}$ at the origin in the sense that for $t>0$, $ |x|^{frac {2} {alpha }} u(t,x) to beta $ as $ |x| to 0$, where $beta = frac {2} {alpha } ( N -2 - frac {2} {alpha } ) $. These solutions in general are neither stationary nor self-similar.
The paper gives a detailed study of long-time dynamics generated by weakly damped wave equations in bounded 3D domains where the damping exponent depends explicitly on time and may change sign. It is shown that in the case when the non-linearity is superlinear, the considered equation remains dissipative if the weighted mean value of the dissipation rate remains positive and that the conditions of this type are not sufficient in the linear case. Two principally different cases are considered. In the case when this mean is uniform (which corresponds to deterministic dissipation rates), it is shown that the considered system possesses smooth uniform attractors as well as non-autonomous exponential attractors. In the case where the mean is not uniform (which corresponds to the random dissipation rate, for instance, when this dissipation rate is generated by the Bernoulli process), the tempered random attractor is constructed. In contrast to the usual situation, this random attractor is expected to have infinite Hausdorff and fractal dimension. The simplified model example which demonstrates infinite-dimensionality of the random attractor is also presented.
We consider the nonlinear heat equation $u_t - Delta u = |u|^alpha u$ on ${mathbb R}^N$, where $alpha >0$ and $Nge 1$. We prove that in the range $0 < alpha <frac {4} {N-2}$, for every $mu >0$, there exist infinitely many sign-changing, self-similar solutions to the Cauchy problem with initial value $u_0 (x)= mu |x|^{-frac {2} {alpha }}$. The construction is based on the analysis of the related inverted profile equation. In particular, we construct (sign-changing) self-similar solutions for positive initial values for which it is known that there does not exist any local, nonnegative solution.
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