Infinite-time gradient blow-up in a degenerate parabolic equation
Summary: This work deals with the Dirichlet problem for the degenerate parabolic equation \(u_t = u^pu_{xx} + u^q\) in a bounded interval \(\Omega \subset \mathbb R\). It is shown that whenever the initial data \(u_0\) belong to \(W^{1,\infty} (\Omega )\), are nonnegative and vanish on \(\partial \Omega\) , the so-called maximal solution \(u\) undergoes an infinite-time gradient blow-up. That is, the function \(u(\cdot, t)\) belongs to \(W^{1,\infty} (\Omega )\), for all \(t \in [0,\infty )\), but we have \(\| u_x(\cdot, t)\|_{L^{\infty} (\Omega )} \to \infty\) as \(t \to \infty\). Moreover, it is shown that if \(q< p-1\) then for sufficiently large \(m > 1\), even the functional \(\int_{\Omega} u^{\alpha} |u_x|^m\) blows up for some \(\alpha = \alpha (m) \geq 0\). Finally, by providing explicit upper estimates for the growth of \(u_x\) with respect to time, it is shown that the rate of gradient blow-up in any of the integral norms considered above is not faster than algebraic, provided that \(q > 1\). In the special case when \(u_0(x) \geq c\,\text{dist}(x, \partial \Omega )\) for all \(x \in \Omega\) and some \(c > 0\), the same is valid for the norm of \(u_x\) in \(L^{\infty} (\Omega )\).
scientific article; zbMATH DE number 6108267