论文标题
关于具有非线性不均匀梯度和源项的自由边界系统的弓和全球径向解决方案的表征
On the characterization of Bow-up and Global Radial solution for free boundary system with nonlinear inhomogeneous gradient and source term
论文作者
论文摘要
本文涉及由\ begin {align} \ label {bs_pr} \ tag {1.1} \ left \ left \ oken {arnay {array} {rl} {rl} {rl} {rl} {rl} {rl} u_t（t，r）=Δu（t，r）=Δu（t，t，t，t，t，t，t，t，t，t，t，t，t，t，t，t，t，t，t， u（t，r）|^α + a（t，x）v^{p}（t，t，r），＆t> 0，\ 0 <r <h（t），\\ v_t（t，r）=ΔV（t，t，r）-λ（t，r）-λ（t，t，t，x，x，x，x）| \ nabla v（t，t，t，t，t，t，r）|^α + a（t，t，t，t，t，t） <r <g（t），\ end {array} \ right。 \ end {align}其中$ r = | x |，\ x \ in \ r^n $，$ p，α> 1 $被赋予常数，$λ（t，x），a（t，x）$满足合适的规定增长条件。首先，我们将本地解决方案的良好性显示为（\ ref {bs_pr}）。其次，我们通过建立$α$，$ p $和系数增长率之间的某些关系来成功对爆炸和全球现象进行分类，在这种关系中，基于Stampacchia截断方法的比较原理证明了比较原则起着核心作用。特别是，如果$ 1 <α<p $和$（u（0，r）; v（0，r））= a（ϕ（r）;φ（r））$是（\ ref ref {bs_pr}）的初始数据，我们发现两个某些正阈值$ a^*_ g $ and $ a^*_ b $ $ a^*_ g $ $ 0的$ 0 $ 0 <a^a <A解决方案的适当值为$ a $，因此$ a^*_ g \ leq a <a^*_ {b} $，而blow-up solutions则为$ a \ geq a \ geq a^{*} _ {b} $保留。另一方面，如果$α\ geq p $结合了$β，p $和$α$的适当比较，则存在全球解决方案，具有指数衰减的非负初始数据。我们的方法与著名的作品\ Cite {MF，PS}有很大不同，在该作者只能以恒定系数处理方程。据我们所知，这是揭示不均匀系数对具有非线性梯度和不同自由界限的合作系统的爆炸和全球现象的影响的第一项工作。
This paper concerns the characterization of blowup and global radial solutions of a two-free boundaries system read by \begin{align}\label{bs_pr} \tag{1.1} \left\{\begin{array}{rl} u_t(t,r)= Δu(t,r) - λ(t,x)|\nabla u(t,r)|^α + a(t,x)v^{p}(t,r),& t>0,\ 0<r<h(t),\\ v_t(t,r) = Δv(t,r) - λ(t,x) |\nabla v(t,r)|^α+ a(t,x)u^{p}(t,r), & t>0,\ 0 < r < g(t), \end{array}\right. \end{align} where $r = |x|,\ x \in \R^N$, $p, α>1 $ are given constants and $λ(t,x), a(t,x)$ satisfy suitable prescribed growth conditions. First, we show the well-posedness of the local solution to (\ref{bs_pr}). Second, we succeed to classify the blowup and global phenomena by establishing some relations between $α$, $p$ and growth rate of the coefficients, in which proving a comparison principle based on the Stampacchia truncation method plays the central role. In particular, if $1<α< p$ and $ (u(0,r);v(0,r)) = A (ϕ(r); φ(r))$ is an initial data of (\ref{bs_pr}), we find two certain positive thresholds $A^*_G$ and $A^*_B$ such that the global fast solution exists for $0 < A < A_G^{*}$ and the global slow solution exists for a suitable value of $A$ such that $A^*_G \leq A < A^*_{B}$ while blow-up solutions hold for $A \geq A^{*}_{B}$. On the other hand, if $α\geq p$ incorporating with a suitable comparison on $β, p$ and $α$, then there exist global solutions with nonnegative initial data of exponential decay. Our approach is being far different from the celebrated works \cite{MF,PS}, where the authors can only handled the equations with constant coefficients. To our knowledge, this is the first work revealing the influence of the inhomogeneous coefficients to the blow-up and global phenomena to the cooperative system with nonlinear gradient and different free boundaries.
