Gradient blowup rate for a semilinear parabolic equation
Zhengce Zhang, Bei Hu · Discrete and Continuous Dynamical Systems · 2009
We present a one-dimensional semilinear parabolic equation$u_t=$u xx$ +x^m |u_x|^p, p> 0, m\geq 0$, forwhich the spatial derivative of solutions becomes unbounded in finite time while the solutions themselvesremain bounded. We show that the spatial derivative of solutions is globally bounded in the case$p\leq m+2$ while blowup occurs at the boundary when $p>m+2$. Blowup rate is also found for somerange of $p$.