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Let $u$ be the solution of $u_t=Deltalog u$ in $R^Ntimes (0,T)$, N=3 or $Nge 5$, with initial value $u_0$ satisfying $B_{k_1}(x,0)le u_0le B_{k_2}(x,0)$ for some constants $k_1>k_2>0$ where $B_k(x,t) =2(N-2)(T-t)_+^{N/(N-2)}/(k+(T-t)_+^{2/(N-2)}|x|^2)$ is the Barenblatt solution for the equation. We prove that the rescaled function $4{u}(x,s)=(T-t)^{-N/(N-2)}u(x/(T-t)^{-1/(N-2)},t)$, $s=-log (T-t)$, converges uniformly on $R^N$ to the rescaled Barenblatt solution $4{B}_{k_0}(x)=2(N-2)/(k_0+|x|^2)$ for some $k_0>0$ as $stoinfty$. We also obtain convergence of the rescaled solution $4{u}(x,s)$ as $stoinfty$ when the initial data satisfies $0le u_0(x)le B_{k_0}(x,0)$ in $R^N$ and $|u_0(x)-B_{k_0}(x,0)|le f(|x|)in L^1(R^N)$ for some constant $k_0>0$ and some radially symmetric function $f$.
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Let $nge 3$ and $psi_{lambda_0}$ be the radially symmetric solution of $Deltalogpsi+2betapsi+beta xcdot ablapsi=0$ in $R^n$, $psi(0)=lambda_0$, for some constants $lambda_0>0$, $beta>0$. Suppose $u_0ge 0$ satisfies $u_0-psi_{lambda_0}in L^1(R^n)$ and $u_0(x)approxfrac{2(n-2)}{beta}frac{log |x|}{|x|^2}$ as $|x|toinfty$. We prove that the rescaled solution $widetilde{u}(x,t)=e^{2beta t}u(e^{beta t}x,t)$ of the maximal global solution $u$ of the equation $u_t=Deltalog u$ in $R^ntimes (0,infty)$, $u(x,0)=u_0(x)$ in $R^n$, converges uniformly on every compact subset of $R^n$ and in $L^1(R^n)$ to $psi_{lambda_0}$ as $ttoinfty$. Moreover $|widetilde{u}(cdot,t)-psi_{lambda_0}|_{L^1(R^n)} le e^{-(n-2)beta t}|u_0-psi_{lambda_0}|_{L^1(R^n)}$ for all $tge 0$.
We introduce a new model of the logarithmic type of wave-like equation with a nonlocal logarithmic damping mechanism, which is rather weakly effective as compared with frequently studied fractional damping cases. We consider the Cauchy problem for this new model in the whole space, and study the asymptotic profile and optimal decay and/or blowup rates of solutions as time goes to infinity in L^{2}-sense. The operator L considered in this paper was used to dissipate the solutions of the wave equation in the paper studied by Charao-Ikehata in 2020, and in the low frequency parameters the principal part of the equation and the damping term is rather weakly effective than those of well-studied power type operators.
We introduce a new model of the logarithmic type of wave like plate equation with a nonlocal logarithmic damping mechanism. We consider the Cauchy problem for this new model in the whole space, and study the asymptotic profile and optimal decay rates of solutions as time goes to infinity in L^{2}-sense. The operator L considered in this paper was first introduced to dissipate the solutions of the wave equation in the paper studied by Charao-Ikehata in 2020. We will discuss the asymptotic property of the solution as time goes to infinity to our Cauchy problem, and in particular, we classify the property of the solutions into three parts from the viewpoint of regularity of the initial data, that is, diffusion-like, wave-like, and both of them.
Let $nge 3$, $0<m<frac{n-2}{n}$, $rho_1>0$, $betagefrac{mrho_1}{n-2-nm}$ and $alpha=frac{2beta+rho_1}{1-m}$. For any $lambda>0$, we will prove the existence and uniqueness (for $betagefrac{rho_1}{n-2-nm}$) of radially symmetric singular solution $g_{lambda}in C^{infty}(R^nsetminus{0})$ of the elliptic equation $Delta v^m+alpha v+beta xcdot abla v=0$, $v>0$, in $R^nsetminus{0}$, satisfying $displaystylelim_{|x|to 0}|x|^{alpha/beta}g_{lambda}(x)=lambda^{-frac{rho_1}{(1-m)beta}}$. When $beta$ is sufficiently large, we prove the higher order asymptotic behaviour of radially symmetric solutions of the above elliptic equation as $|x|toinfty$. We also obtain an inversion formula for the radially symmetric solution of the above equation. As a consequence we will prove the extinction behaviour of the solution $u$ of the fast diffusion equation $u_t=Delta u^m$ in $R^ntimes (0,T)$ near the extinction time $T>0$.
In the framework of the nonsmooth critical point theory for lower semi-continuous functionals, we propose a direct variational approach to investigate the existence of infinitely many weak solutions for a class of semi-linear elliptic equations with logarithmic nonlinearity arising in physically relevant situations. Furthermore, we prove that there exists a unique positive solution which is radially symmetric and nondegenerate.
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