Extinction profile of solutions of the logarithmic diffusion equation on $\mathbb{R}$ (Analysis on Shapes of Solutions to Partial Differential Equations)

全文

(1)92. 数理解析研究所講究録第2046巻 2017年 92-103. Extinction. profile. of solutions of the. diffusion equation. logarithmic. \mathbb{R}. on. Eiji Yanagida Department of Mathematics Tokyo Institute of Technology Introduction. 1. This article is based. versity). on a. and Peter Takáč. joint. work with Masahiko. (Rostock University).. of positive solutions for the. Shimojo (Oakayama Science. Our aim is to investigate the behavior. logarithmic diffusion equation. u_{t}=(\log u)_{xx}, x\in \mathbb{R} In. particular,. Before. we are. discussing. Uni‐. (1.1). .. interested in the behavior of solutions which vanish in finite time.. the above equation,. we. describe. our. motivation to. study. the above. equation.. Equation (1.1). is. a. special. case. of. a more. general. nonlinear diffusion equation of the. form. u_{t}=\nabla\cdot(u^{m-1}\nabla u) , x\in \mathbb{R}^{N} which has been studied linear heat. extensively. (1.2). ,. in the last few decades.. When. m. 1,. =. (1.2). is. a. equation.. u_{t}= $\Delta$ u, x\in \mathbb{R}^{N}. It is not hard to prove. by using. L^{1} ‐solution behaves like. a. the. representation formula of solutions that. fundamental solution. (Gaussian kernel). u(x, t)\displaystyle \sim\frac{C}{t^{N/2} \exp(-\frac{|x|^{2} {4t}) (t\simeq\infty) where C. >. 0 is. a. constant. depending. on. initial data.. for. any. large. positive. t>0 :. .. More. precisely,. satisfies. t^{N/2}u(t^{1/2}x, t)\rightarrow C\exp(-|x|^{2}/4) (t\rightarrow\infty). ,. the solution.

(2) 93. which. gives. asymptotic rescaled profile of the solution.. an. When m>1 ,. (1.2). is small if u>0 is. is called the porous media. small, there. support. An example of such. a. may exist. a. rate. u^{m-1}. boundary. of its. equation. Since the diffusion. solution with. natural free. a. solution is the Barenblatt solutions. (forward. self‐similar. solutions) given by. u(x, t)=t^{- $\alpha$} $\varphi$(t^{- $\beta$}x). ,. where. $\alpha$=\displaystyle \frac{N}{2+N(m-1)}, $\beta$=\frac{1}{2+N(m-1)},. $\varphi$(y)=\displaystyle \{C-\frac{(m-1) $\beta$}{2}|y|^{2}\}_{+}^{1/(m-1)}. It is known that for any to. one. compactly supported. of the Barenblatt solutions with the. initial. same. data, the solution of (1.2). total mass, that. t^{ $\alpha$}u(t^{ $\beta$}x, t)\rightarrow $\varphi$(x) (t\rightarrow\infty) In other. words,. the linear heat. the Barenblatt solution. equation. When m<1 ,. (1.2). in. describing. is called. a. plays. the. singular. because the diffusion rate u^{m-1} tends to. depends. on m. the. same. converges. is,. .. role. as. the Gaussian solution for. asymptotic rescaled profile. diffusion. \infty as. equation. u\rightarrow 0. .. or a. fast diffusion. equation,. The behavior of solutions. crucially. and N , and the exponent. m_{c}:=\displaystyle \frac{N-2}{N} (N\geq 3) turns out to be critical.. for any positive initial for all t>0. .. If N \geq 3 and. data,. m. > m_{\mathrm{c} ,. the solution exists. diffusion is not fast. globally. enough. in time and remains. It is known that there exist forward self‐similar solutions. so. that. positive. (Barenblatt‐like. solutions) given by. u(x, t)=t^{- $\alpha$} $\varphi$(t^{- $\beta$}|x|) (t>0) where. $\alpha$=\displaystyle \frac{N}{Nm-(N-2)}>0, $\beta$=\frac{1}{Nm-(N-2)},. It. was. any. shown. $\varphi$(y):=\displaystyle \{C+\frac{(1-m) $\beta$}{2}|y|^{2}\}^{-1/(1-m)}. by Vázquez [29], Carrillo. positive L^{1} solution. total. converges to. et al.. one. [7, 8, 9], Doskalopoulos‐Sesum [10]. of the Barenblatt‐like solution with the. mass:. t^{ $\alpha$}u(t^{ $\beta$}x, t)\rightarrow $\varphi$(x) (t\rightarrow\infty). .. that same.

(3) 94. On the other that. so. if N<3. hand,. or. if N\geq 3 ànd m<m_{c} , then the diffusion is very fast. finite time extinction of solutions may. a. initial value is positive and. decay. to 0. as. |x|. [17, 18, 29].. occur. \rightarrow\infty ,. More. if the. precisely,. then there exists T<\infty such that. the solution satisfies. \displaystyle \lim_{t\uparrow T}u(x, t)=0 uniformly. in x\in \mathbb{R}. (backward. .. An. Barenblatt. example of such. a. solution is the backward self‐similar solutions. solutions) gien by. u(x, t)=(T-t)^{ $\alpha$} $\varphi$((T-t)^{ $\beta$}x) , t\in(-\infty, T). ,. where. $\alpha$=\displaystyle \frac{N}{N(1-m)-2}>0, $\beta$=\frac{1}{N(1-m)-2}>0,. and. It. was. $\varphi$(y)=\displaystyle \{C+\frac{ $\beta$(1-m)}{2}|y|^{2}\}^{-1/(1-m)}. shown. by Blanchet. class of initial. data,. [2],. et al.. Bonforte et al.. the solution converges to. one. [3],. us. consider the. that for. a. .. where m=1 :. case. u_{t}= $\Delta$(\log u) This. wide. of the backward Barenblatt solutions:. (T-t)^{- $\alpha$}u((T-t)^{- $\beta$}x, t)\rightarrow $\varphi$(x) (t\uparrow T) Now let. [5, 6]. Fila et al.. equation has been studied. (1.3). in various contexts.. It appears. as. the central limit. approximation of Carlemans model of the Boltzman equation [23], and the expansion into. a vacuum. of. a. thermalized electron cloud. Van‐der‐Waals interactions in thin films of a nonlinear fourth order terms. are. [21].. It also arises. liquid spreading. neglected.. There is. a. given by \partial_{ $\tau$}g_{ij}. g_{ij}=w$\delta$_{ij} to. (1.3). For. the. ,. where. -2R_{ $\eta$ j}. $\delta$_{ij}. .. If the metric is. general. a. analysis of (1.3),. see. there is. a. as. above,. on. g_{ij}( $\tau$). w. such that. problem. is reduced. function. [10, 11, 19, 20, 24, 26, 27, 28].. the dimension N\geq 3 is the very fast diffusion. results described. surface,. long. if certain. Riemannian metric. denotes the standard Euclidean metric, and the. For mathematical. (1.3),. conformal,. solid. model for. relation with the Ricci flow. \mathbb{R}^{2} in differential geometry [13, 16], namely the evolution of =. on a. as a. case.. This. case. is covered. by. and the backward self‐similar solution is written. as. u(x, t)=(T-t)^{N/(N-2)} $\varphi$((T-t)^{1/(N-2)}x) t\in(-\infty, T). ,.

(4) 95. where T>0 is. an. extinction time and. $\varphi$(y)=\displaystyle \{C+\frac{1}{2(N-2)}|y|^{2}\}^{-1} and any. positive solution. converges to. one. of the self‐similar solutions:. (T-t)^{-N/(N-2)}u((T-t)^{-1/(N-2)}x, t)\rightarrow $\varphi$(x) (t\uparrow T) If N=2 ,. by setting. u(x, t) :=(T-t) $\varphi$(x) we. .. ,. have. u_{t}=- $\varphi$, $\Delta$(\log u)= $\Delta$(\log $\varphi$) Hence. u. satisfies the. logarithmic. .. diffusion equation, if. $\Delta$(\log $\varphi$)+ $\varphi$=0. This. equation has. a. satisfies the Gelfand value. positive radial. equation. u(x, 0)=u_{0}(x). is. entire solution for N. $\Delta$ w+e^{w}=0 It .. was. shown. ,. where $\alpha$,. $\beta$. [15]. studied the. given positive. The initial value u_{0}. :. (1.4) [0, T). in the classical. [14]. \displaystyle \lim_{x\rightar ow+\infty}(\log u)_{x}=- $\beta$,. as. (1.4). t>0 ,. corresponding to the decay rate of. u as. x\rightarrow\pm\infty.. at t=0 is assumed to be in the function space. possesses. at least for small. a. at x=\pm\infty. .. Then it. was. L^{1}(\mathbb{R}). shown in. [14]. unique positive solution which satisfies. t>0. .. Here and in what. follows,. the interval. positive classical. (1.4).. Concerning proved. t\rightarrow\infty , then. .. will denote the maximal time interval for the existence of the. solution to. \log $\varphi$. that if the initial. x\in \mathbb{R},. problem (1.4). sense. =. x\in \mathbb{R}, t>0,. constants. \mathbb{R}\rightarrow(0, \infty). w. problem. incorporating prescribed asymptotic conditions that the initial value. Note that. positive, radially symmetric and u_{0}(x)\rightarrow 0. \left{bginary}{l \primeu_{t}=(\logu)_{x},\ lim_{x\rghtaow-\infty}(logu)_{x}=$\alph$,\ u(x,0)=_{}(x), \end{ary}\ight.. are. .. Hsu. by. (T-t)^{-1}u(x,t)\rightarrow $\varphi$(x) (t\uparrow T) For N=1 Hsu. 2. =. that if. the behavior of solutions of. $\alpha$= $\beta$. and. u_{0}(x). is. even. (1.4). near. the extinction time, Hsu. symmetric, then. (T-t)^{-1}u(x, t)\rightarrow $\varphi$(x) (t\uparrow T). ,. [15].

(5) 96. where $\varphi$ is. a. positive solution of. \left{\begin{ar y}{l (\log$\varphi$)_{x}+$\varphi$=0,x\in mathb{R},\ frac{$\varphi$_{x} $\varphi$}\rghtarow$\alph$>0(x\rightarow\pminfty). \end{ar y}\right. The aim of this article is to. study. the. case. where. Our main result states that the rescaled solution a. traveling pulse. $\alpha$\neq $\beta$. or. (1.5) u_{0}(x). not. even. v(x, s) :=e^{s}u(x, T-e^{-s}). symmetric.. converges to. solution of. v_{s}=(\log v)_{xx}+v, x\in \mathbb{R}. Theorem 1 Let. u. be. a. solution. of (1.4).. There exists. $\gamma$\in \mathbb{R} such. that. v(x, t)=e^{s}u(x, T-e^{-s})\rightarrow $\varphi$(x-cs- $\gamma$) (s\rightarrow\infty) uniformly. in. x\in \mathbb{R}_{f} where. satisfies. $\varphi$. -c$\varphi$'=(\log $\varphi$ + $\varphi$, z\in \mathbb{R}, with the. asymptotic conditions. \displayst le\lim_{z\rightarow-\infty}\frac{$\varphi$'(z)}{$\varphi$(z)}=$\alpha$, \displaystyle\lim_{z\rightar ow+\infty}\frac{$\varphi$'(z)}{$\varphi$(z)}=-$\beta$. The rest of this paper is damental a. properties of solutions of. transformation which is useful in. consider the existence of we. 2. organized. sketch the. proof. equation. in. we. (1.4). that the total. the. follows. In Section 2. we. describe. logarithmic diffusion equation, and. studying the. traveling solutions. some. fun‐. introduce. extinction behavior. In Section 3. of the transformed. equation.. we. In Section 4. of Theorem 1.. Fundamental. In this section. as. properties. describe fundamental over. \mathbb{R} and. applying. properties of solutions the. to. asymptotic conditions. mass. m(t) :=\displaystyle \int_{\mathb {R} u(x, t)dx satisfies. \displaystyle \frac{d}{dt}m(t)=-( $\alpha$+ $\beta$). .. (1.4). Integrating at. x. =. \pm\infty ,. the. we see.

(6) 97. Hence the solution vanishes at. t=T :=\displaystyle \frac{1}{ $\alpha$+ $\beta$}\int_{\mathb {R} u_{0}(x)dx Introducing. (2.1). .. the transformation of variables. u(x, t)=(T-t)v(x, s) , t=T-e^{-s}, we. have. Thus,. in order to. \left{bginary}{l v_s=(\logv)_{x}+,\inmathb{R},s\in(-logT,\mathr{o}\mathr{o})\ (logv)_{x}\rightaow+$\alph>0(x\rightaow-\infty) (\logv)_{x}\rightaow-$\beta<0(x\rightaow+\infty) v(x,-\logT)=v_{0}(x:=\frac{1}Tu_0(x) \end{ary}\ight.. study. the behavior of. u as. t\rightarrow T ,. we. need to. (2.2). study. the behavior of. v. as S \rightarrow \infty.. Integrating. the. equation. in. (2.2). and using the asymptotic conditions,. \displaystyle \frac{d}{ds}\int_{\mathb {R} v(x, s)dx=\int_{\mathb {R} v(x, s)dx-( $\alpha$+ $\beta$). we. have. .. Hence. \displaystyle\int_{\mathb {R} v_{0}(x)dx On the other. \{. > $\alpha$+ $\beta$. \Rightarrow. total. mass. grows. = $\alpha$+ $\beta$. \Rightarrow. total. mass. is conserved. < $\alpha$+ $\beta$. \Rightarrow. total. mass. vanishes in finite time. exponentially. hand, by (2.1) and. v(x, -\displaystyle \log T)=v_{0}(x)=\frac{1}{T}v_{ $\Phi$}(x) we. ,. have. \displaystyle \int_{\mathb {R} v_{0}(x)dx= $\alpha$+ $\beta$. Hence the total. 3. mass. of. v. is conserved.. Traveling solutions. When the. mass. is conserved in. (2.2),. stationary solution. Indeed, when and the solution converges to the. $\alpha$. one =. may. expect that the solution converges. $\beta$ problem (1.4) ,. stationary solution. as. admits. a. to. a. stationary solution. is discussed in. [10, 14]. However,.

(7) 98. when. $\alpha$\neq $\beta$. must be. the. possibility. one. ,. v(x, s) = $\varphi$(z+ $\gamma$) of the. we see. positive stationary solutions, because. no. symmetric with respect. $\alpha$\neq $\beta$. profile. (1.4),. there exist. even. When form. ,. traveling. that $\varphi$ must. wave. where. a. traveling. denotes the. c. and $\gamma$ \in \mathbb{R} is. a. stationary solution. point.. is the convergence to. z=x-cs ,. ,. to its critical. any. wave. solution of the. propagation speed,. $\varphi$ > 0 is. phase shift. Substituting v= $\varphi$(z). satisfy. \left\{ begin{ar y}{l (\log$\varphi$)_{z }+c$\varphi$_{z}+$\varphi$=0,z\inR\ (\log$\varphi$)_{z}(-\infty)=$\alpha$,(\log$\varphi$)_{z}(-\infty)=-$\beta$. \end{ar y}\right. In order to fix the. in. phase,. we. may. (3.1). require. $\varphi$_{z}(0)=0. The unique existence of such. c_{<}\geqq 0 if and only if $\alpha$_{<}\geq $\beta$ as a. traveling. If. wave. introduce. we. .. a. traveling. Therefore,. wave. for. will be shown. $\alpha$= $\beta$. the. ,. later, and. stationary solution. it turns out that can. be. regarded. solution with c=0. an. auxiliary. variable. $\psi$. :=. (\log $\varphi$)_{z}. ,. then. (3.1). is. equivalent. to the. system. the. By c. =. phase plane analysis,. c( $\alpha$, $\beta$). speed. of the. such that. (3.2). traveling. wave. \left{\begin{ar y}{l $\varphi$_{z}=$\varphi\psi$,\ $\psi$_{z}=-c$\varphi\psi$- \varphi$. \end{ar y}\right. we can. has. an. (3.2). show that for each $\alpha$,. ,. there exists. connecting (0, $\alpha$) Moreover,. orbit. is monotone. $\beta$>0. decreasing. in. $\alpha$. the. a. unique. propagation. and monotone increasing in. $\beta$,. and satisfies satisfies. c\geq 0<\Leftrightar ow $\alpha$<\geq $\beta$ Stability linearized. of the. traveling. wave. solution. can. be studied. by considering. the. following.. eigenvalue problem:. $\lambda$ U=(U/ $\varphi$)_{zz}+cU_{z}+U, z\in \mathbb{R}, (U/ $\varphi$)_{z}\rightarrow 0 It is easy to. see. that. $\lambda$_{0}. =. 1 is. by U_{0}= $\varphi$+c$\varphi$_{z} Similarly, $\lambda$_{1} .. U_{0}=$\varphi$_{z}. ,. which. corresponds. an. is unstable in the linearized. an. with the associated. eigenvalue. spatial phase. sense.. (3.3). z\rightarrow\pm\infty.. eigenvalue. =0 is. to the. as. with the associated. shift. Thus the. traveling. However, if the disturbance. conservation, then the traveling solution. eigenfunction given. may be stable.. is in. a. eigenfunction wave. solution. class of. mass.

(8) 99. Outline of the. 4 We. the. study. STEP 1:. case. of critical mass, and prove the convergence to. points of. two distinct solutions is. The intersection number. The. Theorem 1. We first establish the intersection number. intersection. cause we. proof of. must exclude the. parabolic. STEP 2:. possibility of. We show the boundedness of. if the. if the. STEP 3:. mass. is. mass. the number of. s.. is not trivial for the fast diffusion. principle. appearance of intersection. In. v. and v_{x}. fact,. by. need. we. equation, be‐. points from. a more. \pm\infty.. precise result. the. comparison argument. We also. the intersection number argument.. by. It should be noted that this property holds truw. fact,. in. solution.. words. need lower estimates. In. traveling. principle, namely,. non‐increasing. comparison principle follows immediately.. about. a. is smaller than. $\alpha$+ $\beta$. larger than $\alpha$+ $\beta$ then ,. ,. only in the case of mass conservation.. then the solution vanishes in finite. the solution grows. time, whereas. exponentially.. We show that the solution becomes unimodal in finite time.. To show. this,. boundedness of. v. we use. the fact that \mathrm{t}v=e^{s} satisfies the. and the intersection number. equation. principle, the solution. in. (2.2). By. the. must be unimodal. eventually. STEP 4:. We show that the solution becomes. To show. this,. we use. equation. in. in finite time.. the fact that. v(x, s)=\exp(Px+Q+s) satisfies the. \log‐concave. (2.2).. ( P, Q : consttans). Then the intersection number. principle implies that there. exists s_{0}<\infty such that. (\log v)_{xx}<0 Once the solution becomes preserves the. \log‐concavity.. \log‐concave,. for x\in \mathbb{R}, s>s_{0}. then the maximum. principle implies. that it.

(9) 100. STEP 5:. We derive. If the solution is. allows x. .. If. to. us. we can. an. equation of the. evolution. the flux $\theta$. \log‐concave,. curvature of. :=-(\log v)_{x}. is. v. in terms v_{x}.. increasing. in x\in \mathbb{R} , which. parametrize the solution by thê flux density $\theta$ instead of the take. $\theta$\in(- $\alpha$, $\beta$). independent. as a new. space. variable,. space variable. the curvature. k(x, s)=-\displaystyle \frac{(\log v)_{xx}}{v}>0 as a. function of. ( $\theta$, s). is described. as. \left{\begin{ar y}{l k_{s}=k^{2}(vk_{$\thea$})_{$\thea$}+k(-1),\ k_{$\thea$}=\frac{1-k}$\thea$}, \thea$=- \alph$, \beta$, \end{ar y}\right. where. (4.1). v=-\displayst le\int_{-$\alpha$}^{$\theta$}\frac{$\theta$}{kd$\theta$.. STEP 6:. We. analyze. convergence of the. Using. a. the evolution. profile of. suitable. v. equation of the. curvature of solutions to show the. to $\varphi$.. Lyapunov function. \displaystyle\frac{d}{ds}\int_{-$\alpha$}^{$\beta$}(\frac{kv}{2}k_{$\theta$}^{2}-\frac{1}{2}k^{2}+k)d$\theta$=-\int_{-$\psi$_{+}^{$\psi$-}\frac{k_{s}^{2}{k}d$\theta$<0, we can. to to. a. apply. the standard argument for. stationary solution of (4.1). traveling. wave. in the. as s\rightarrow\infty. solution is. given by. STEP 7:. We show the convergence to. traveling. the. by. two. This proves the convergence of. crossing property. more. stationary solution corresponds that the. means. to. a. asymptotic profile of the. a. traveling pulse.. (2.2). to. one. of the. traveling. wave. is shown. by. crossing property.. We squeeze the solution. For. to show that k converges. wave.. The convergence of the solution of. analyzing. This. .. which. original equation,. the. dynamical systems. to. v. between the two. details of the. proof,. traveling pulses $\varphi$(x-cs-$\gamma$^{1}) a. $\varphi$(x-cs-$\gamma$^{+}). traveling pulse $\varphi$(x-cs- $\gamma$) by observing. traveling. we. and. solutions. refer the reader to. [25].. .. the.

(10) 101. References [1]. S.. Angenent,. Math. 380. [2]. A.. The. zero. (1988),. Blanchet,. M.. Bonforte,. equation. Bonforte,. J.. Bonforte,. totics. 196. [5]. M.. [7]. M.. (2010),. Fila,. Fila,. and J.L.. Vázquez, Special. Vázquez,. M.. A continuum of extinction rates for the fast. Carrillo,. A.. Jüngel,. methods for. J.. P. A.. Daskalopoulos. ematics. [11]. J.‐S.. Vázquez,. Equations. 1129‐1147.. rate for. Markowich, G. Toscani,. degenerate parabolic problems. P. A.. Carrillo, C. Lederman,. J.L.. (2011),. Dyn. Differ. Equations 17. (2001),. Monatsh. Math. 133. Carrillo,. 10. Yanagida, Convergence. J.A.. P.. fast diffusion with slow asymp‐. Winkler,. M. Winkler and E.. Partial Differ.. [10]. of. Riemannian manifold. Arch. Rat. Mech. Anal.. supercritical nonlinearity.. J.A.. decay. on a. Fine. 28. and C. E.. a. parabolic equation. (2005),. 249‐269.. A.. Unterreiter, Entropy. and. generalized Sobolev. 1‐82.. Markowich, G. Toscani,. linearizations of very fast diffusion. [9]. rates of. equation via functional inequalities. Proc.. equation. Comm. Pure Appl. Anal.. inequalities.. Vázquez, Sharp. 16459‐16464.. J G. Grillo and J.L.. with. J.A.. Angew.. 631‐680.. J.L.. dissipation. [8]. (2010),. Entropy method and flow. diffusion. [6]. J. Reine. entropy estimates. Arch. Rat. Mech. Anal. 191,. via. Dolbeault, G. Grillo. Nat. Acad. Sciences 107 M.. parabolic equation,. Dolbeault, G. Grillo and J.L. Vázquez, Asymptotics. J.. solutions to the nonlinear fast diffusion. [4]. a. (2009). 347‐385 M.. solution of. a. 79‐96.. of the fast diffusion. [3]. set of. Poincaré. equations. Nonlinearity. 15. inequalities. (2002),. for. 565‐580.. asymptotics for fast diffusion equations, Commun.. (2003),. 1023‐1056.. Kenig Degenerate Diffusions,. EMS Tracts in Math‐. 1, European Mathematical Society, 2007.. Guo,. On the. Differential. Cauchy problem for. Equations. 21. (1996),. a. very fast diffusion. 1349‐1365.. equation, Comm. Partial.

(11) 102. [12]. J.‐S.. Guo,. H.. Matano, M. Shimojo and C.‐H. Wu, On. for the curvature flow with. driving force,. free. a. boundary problem. Arch. Ration. Mech. Anal. 219. (2016),. 1207‐1272.. [13]. R.. [14]. S.‐Y. Hsu, Dynamics. [15]. Hamilton,. The Ricci flow. surfaces, Contemp.. extinction time of. near. Ann. 323. (2002),. S.‐Y.. Behaviour of solutions ofa. Hsu,. S.‐Y.. Hsu,. (2004),. 56. K. M. Hui and S.. Manuscripta. [19]. H. G.. Kaper. H. G.. K. E.. Longman. (1976), [22]. H.. H. P.. 21. [24]. A.. near. the extinction. diffusion equation with bounded. (2013), of the. Edinburgh. 103‐111.. higher‐dimensional logarithmic. (2013),. Sect. A 143. 817‐830.. Initial value. problems for the Carleman equation,. 343‐362.. K. Leaf and S.. and A.. 491‐524.. Reich, Convergence. equation,. Math. Methods. Hirose, Expansion. of. an. of. Appl.. semigroups Sci. 2. electron. with. an. appli‐. (1980), 303‐308.,. cloud, Phys.. Lett. A 59. 285‐286.. Matano, Nonincrease of the lap‐numUer of a solution for. linear. [23]. Leaf,. (1980),. cation to the Carleman. [21]. (2014),. and G. K.. Kaper, G.. Math.. Kim, Extinction profile of the logarithmic diffusion equation,. Math. 143. Nonlinear Anal. 4. [20]. singular diffusion equation,. logarithmic. Kim, Large‐time behaviour. diffusion equation, Proc. Roy. Soc.. [18]. 237‐262.. 63‐104.. Existence of solution of the. K. M. Hui and S.. (1988),. singular diffusion equation. above Gauss curvature, Nonlinear Anal. 77. [17]. a. Math. 71. 281‐318.. time, Nonlinear Anal.. [16]. on. a. parabolic equation, J. Fac. Sci. Univ. Tokyo Sect. IA. one‐dimensional semi‐ 29. (1982). 401‐441.. McKean, The central limit theorem for Carlemans equation, Israel J. Math.. (1975),. 54‐92.. Rodríguez. and J. L.. Vázquez,. Arch. Rational Mech. Anal. 110. A well. posed problem. (1990),. 141‐163.. in. singular Fickian diffusion,.

(12) 103. [25]. M.. Shimojo,. logarithmic. [26]. P.. P. Takáč and E.. diffusion. P.. equation with. a. a. monotone local semiflow I:. uniqueness, Differential Integral Equations. Takáč, A fast diffusion equation which generates. Global existence and asymptotic. of solutions to the. linear source, preprint.. Takáč, A fast diffusion equation which generates. Local existence and. [27]. Yanagida, Asymptotic behavior. a. 4. (1991),. 151‐174.. monotone local semiflow II:. behavior, Differential Integral Equations. 4. (1991),. 175‐187.. [28]. J. L.. Vázquez, Nonexistence. diffusion type, J. Math. Pures. [29]. J. L.. of solutions for nonlinear heat. Appl.. 71. Vázquez, Finite‐time blow‐down. logarithmic diffusion,. Discrete Contin.. (1992),. equations of fast‐. 503‐526.. in the evolution of. Dyn. Syst.. 19. point. (2007),. masses. 1‐35.. by planar.

(13)