The behavior of the free boundary for reaction-diffusion equations with convection in an exterior domain with Neumann or Dirichlet boundary condition

详细信息    查看全文

• 作者：Ross G. Pinsky ^{pinsky@math.technion.ac.il" class="auth_mail" title="E-mail the corresponding author}
• 关键词：35R35 ; 35J61
• 刊名：Journal of Differential Equations
• 出版年：2016
• 出版时间：15 March 2016
• 年：2016
• 卷：260
• 期：6
• 页码：5075-5102
• 全文大小：411 K

文摘

Let be a second order elliptic operator and consider the reaction–diffusion equation with Neumann boundary condition, where 615006592&_mathId=si5.gif&_user=111111111&_pii=S0022039615006592&_rdoc=1&_issn=00220396&md5=7a67f7030f50a7124a9c4d84e2d40dcc" title="Click to view the MathML source">p∈(0,1)$p\in (0,1)$, 615006592&_mathId=si6.gif&_user=111111111&_pii=S0022039615006592&_rdoc=1&_issn=00220396&md5=546ca4d24a974958f0481fde21c2ef50" title="Click to view the MathML source">R>0$R>0$, 615006592&_mathId=si130.gif&_user=111111111&_pii=S0022039615006592&_rdoc=1&_issn=00220396&md5=12288cf41bd95a6a34c1bdd230cb8820" title="Click to view the MathML source">h>0$h>0$ and 615006592&_mathId=si8.gif&_user=111111111&_pii=S0022039615006592&_rdoc=1&_issn=00220396&md5=91f98c43eeeeb20493a6a8408d4ade62" title="Click to view the MathML source">Λ=Λ(r)>0$\mathrm{\Lambda }=\mathrm{\Lambda }(r)>0$. This equation is the radially symmetric case of an equation of the form where is a second order elliptic operator, and where 615006592&_mathId=si12.gif&_user=111111111&_pii=S0022039615006592&_rdoc=1&_issn=00220396&md5=6af39d99329690cdea0b12c7302fb469" title="Click to view the MathML source">d≥2$d\ge 2$, 615006592&_mathId=si130.gif&_user=111111111&_pii=S0022039615006592&_rdoc=1&_issn=00220396&md5=12288cf41bd95a6a34c1bdd230cb8820" title="Click to view the MathML source">h>0$h>0$ is continuous, 615006592&_mathId=si13.gif&_user=111111111&_pii=S0022039615006592&_rdoc=1&_issn=00220396&md5=6e98bc628e0b14c190ba1faaa5a43c00" title="Click to view the MathML source">D⊂R^d$D\subset R^d$ is bounded, and 615006592&_mathId=si14.gif&_user=111111111&_pii=S0022039615006592&_rdoc=1&_issn=00220396&md5=f594340a439f16e760541ada46b681bf">615006592-si14.gif">$\overline{n}$ is the unit inward normal to the domain 615006592&_mathId=si15.gif&_user=111111111&_pii=S0022039615006592&_rdoc=1&_issn=00220396&md5=09a5274d1444202e7477f685b8871053">615006592-si15.gif">${\mathbb{R}}^d-\overline{D}$. Consider also the same equations with the Neumann boundary condition replaced by the Dirichlet boundary condition; namely, 615006592&_mathId=si16.gif&_user=111111111&_pii=S0022039615006592&_rdoc=1&_issn=00220396&md5=0b15bfe8e5bd6acc27a5552edbd5488f" title="Click to view the MathML source">u(R)=h$u(R)=h$ in the radial case and 615006592&_mathId=si17.gif&_user=111111111&_pii=S0022039615006592&_rdoc=1&_issn=00220396&md5=db6204783a39ab6394ed76ed7f47f6c6" title="Click to view the MathML source">u=h$u=h$ on ∂D   in the general case. The solutions to the above equations may possess a free boundary. In the radially symmetric case, if 615006592&_mathId=si18.gif&_user=111111111&_pii=S0022039615006592&_rdoc=1&_issn=00220396&md5=907e744591b9b02c5d2b7fd3121b4b10" title="Click to view the MathML source">r^⁎(h)=inf⁡{r>R:u(r)=0}<∞$r^{⁎}(h)=\inf \{r>R:u(r)=0\}<\infty$, we call this the radius of the free boundary; otherwise there is no free boundary. We normalize the diffusion coefficient A to be on unit order, consider the convection vector field B   to be on order 615006592&_mathId=si19.gif&_user=111111111&_pii=S0022039615006592&_rdoc=1&_issn=00220396&md5=855f5953d8c795a91ba91db410013e7e" title="Click to view the MathML source">r^m$r^m$, 615006592&_mathId=si20.gif&_user=111111111&_pii=S0022039615006592&_rdoc=1&_issn=00220396&md5=a87c73a785beb6b20718b73bd746be82" title="Click to view the MathML source">m∈R$m\in R$, pointing either inward 615006592&_mathId=si21.gif&_user=111111111&_pii=S0022039615006592&_rdoc=1&_issn=00220396&md5=41dc5f0c9c36ef235ce6ea40b0b8012d" title="Click to view the MathML source">(−)$(-)$ or outward 615006592&_mathId=si22.gif&_user=111111111&_pii=S0022039615006592&_rdoc=1&_issn=00220396&md5=1174c095a29d03b79a40d6f47b2fedfb" title="Click to view the MathML source">(+)$(+)$, and consider the reaction coefficient Λ to be on order 615006592&_mathId=si23.gif&_user=111111111&_pii=S0022039615006592&_rdoc=1&_issn=00220396&md5=41537d463cf824f8bc9e8754933017ff" title="Click to view the MathML source">r^−j$r^{-j}$, 615006592&_mathId=si24.gif&_user=111111111&_pii=S0022039615006592&_rdoc=1&_issn=00220396&md5=4aaea48c4dcb0b57f5b397529f82d554" title="Click to view the MathML source">j∈R$j\in R$. For both the Neumann boundary case and the Dirichlet boundary case, we show for which choices of m  , 615006592&_mathId=si25.gif&_user=111111111&_pii=S0022039615006592&_rdoc=1&_issn=00220396&md5=79efb902877714652db77ebf05878708" title="Click to view the MathML source">(±)$(\pm )$ and j a free boundary exists, and when it exists, we obtain its growth rate in h as a function of m  , 615006592&_mathId=si25.gif&_user=111111111&_pii=S0022039615006592&_rdoc=1&_issn=00220396&md5=79efb902877714652db77ebf05878708" title="Click to view the MathML source">(±)$(\pm )$ and j. These results are then used to study the free boundary in the non-radially symmetric case.