Vast Multiplicity of Very Singular Self-Similar Solutions of a Semilinear Higher-Order Diffusion Equation with Time-Dependent Absorption

Journal of Mathematical Sciences

Galaktionov, V. A.¹ 

关键词: The Cauchy problem;    diffusion equations with absorption;    initial Dirac mass;    very singular solutions;    existence;    nonexistence;    bifurcations;    branching.;   

DOI  :  

学科分类：数学（综合）

来源: University of Tokyo * Department of Mathematical Sciences

【 摘 要 】

Asabasicmodel,theCauchyproblemin$enimese_+$forthe$2m$th-ordersemilinearparabolicequationofthediffusion-absorp-tiontype$$u_t=-(-D)^mu-t^a|u|^{p-1}u,quadmbox{with},,p>1,,,a>0,,,,mge2,$$with{emsingular}initialdata$u_0(x)otequiv0$suchthat$u_0(x)=0$forany$xot=0$,isstudied.Theadditionalmultiplier$h(t)=t^ao0$as$to0$intheabsorptiontermplaysaroleofatime-dependentnon-homogeneouspotentialthataffectsthestrengthoftheabsorptionterminthePDE.Existenceandnonexistenceofthecorresponding{emverysingularsolutions}(VSSs)isstudied.For$m=1$and$h(t)equiv1$,firstnonexistenceresultfor$pgep_0=1+frac2N$wasprovedinthecelebratedpaperbyBrezisandFriedmanin1983.ExistenceofVSSsinthecomplementinterval$1

【 授权许可】

Unknown