Spatially monotone homoclinic orbits in nonlinear parabolic equations of super-fast diffusion type
The subject of this article is the question on the existence of smooth homoclinic orbits in nonlinear parabolic equations without any source. More concretely the authors consider positive classical solutions of the nonlinear diffusion equation \[ u_{t}=u^{p}u_{xx}\eqno (1) \] for \(p>2,\) where \(x\) varies over the real line and the time variable \(t\) is allowed to lie in the whole range \((-\infty, \infty).\) After the substitution \(v=u^{1-p}\) it is transformed into the super-fast diffusion equation \(v_{t}=(v^{m-1}v_{x})_{x}\) with parameter \(m=-\frac{1}{p-1}\) located in the interval \((-1,0)\) according to the requirement \(p>2.\) A positive function \(u\in C^{2,1}(\mathbb{R}\times\mathbb{R})\) is a homoclinic orbit of (1) if \(u(\cdot,t)\rightarrow 0\) locally uniformly on \(\mathbb{R}\) at \(t\rightarrow\pm\infty.\) The main result is the following statement: Theorem. Let \(p>2.\) Then there exists a homoclinic orbit of (1). More precisely, (1) possesses a positive classical solution \(u\in C^{\infty}(\mathbb{R}\times\mathbb{R})\) such that for all \(y\in\mathbb{R}\) one can find \(C>0\) such that \(\|u(\cdot,t)\|_{L^{\infty}((-\infty,y))}\leq Ct^{-\frac{1}{p}}(\ln t)^{\frac{2}{p}}\) for all \(t\geq 2\) and \(\|u(\cdot,t)\|_{L^{\infty}((-\infty,y))}\leq C(-t)^{-\frac{1}{p}}\ln(-t)\) for all \(t\leq -2.\) Moreover, this solution can be constructed in such a way that \(u_{x}>0\) holds in \(\mathbb{R}\times\mathbb{R},\) and that \(u(x,t)\rightarrow 0\) as \(x\rightarrow -\infty,\) even uniformly with respect to \(t\in\mathbb{R},\) that is, \(\|u\|_{L^{\infty}((-\infty,y)\times\mathbb{R})}\rightarrow 0\) as \(y\rightarrow -\infty.\)
scientific article; zbMATH DE number 6133900