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Gradient blowup rate for a semilinear parabolic equation

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  • We present a one-dimensional semilinear parabolic equation $u_t=$u xx$ +x^m |u_x|^p, p> 0, m\geq 0$, for which the spatial derivative of solutions becomes unbounded in finite time while the solutions themselves remain 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 some range of $p$.
    Mathematics Subject Classification: Primary: 35K55, 35B40.


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