Global existence and blow-up for weakly coupled degenerate and singular parabolic equations with localized source
In this article the problem \[ \begin{aligned} u_t-(x^\alpha u_x)_x= u^m(x_0(t), t)v^n(x_0(t), t)&\quad \text{in }(0, a)\times (0,T),\\ v_t-(x^\beta {v_x})_x= v^p(x_0(t), t)u^q(x_0(t), t)&\quad \text{in }(0, a)\times (0, T),\\ u(0, t)=u(a, t)=v(0, t)=v(a, t)=0&\quad \text{in }(0, T),\\ u(x, 0)=u_0(x), v(x, 0)=v_0(x)&\quad \text{in }[0, a] \end{aligned} \] is considered. The system describes conduction of heat related to the geometric shape of the body. Here \(x_0(t):\mathbb R^+\to (0, a)\) is Hölder continuous; \(u_0,v_0\in {\mathcal C}^{2+\gamma}([0, a])\), \(\gamma\in (0,1)\), are non-negative and non-trivial functions satisfying compatibility conditions; \(m\), \(n\), \(p\), \(q\) are real positive constants; \(T>0\), \(a>0\), \(\alpha\), \(\beta\in [0, 2)\) are constants. The authors prove that all solutions exist globally if \(m<1\), \(p<1\), \(nq<(m-1)(p-1)\). In the article, it is proved that the solution exists globally for sufficiently small initial value and blows up in finite time for sufficiently large initial value if \(m>1\) or \(p>1\) or \(nq>(m-1)(p-1)\). If the solution blows up at the same time and \(\alpha=\beta\in (0,1)\), \((x^\alpha u_{0x}(x))_x\leq M_1\), \((x^\alpha v_{0x}(x))_x\leq M_2\) in \((0,a)\) for some positive constants \(M_1\) and \(M_2\), the authors prove that the blow-up set is any closed subset of \((0,a)\). The authors also give the blow-up rate.
scientific article; zbMATH DE number 5947338
scientific article