Extinction of solutions of semilinear higher order parabolic equations with degenerate absorption potential

We study the first vanishing time for solutions of the Cauchy–Dirichlet problem for the 2m-order (m ≥ 1) semilinear parabolic equation $${u_t + Lu + a(x) |u|^{q-1}u=0,\,0 < q < 1}$$ with a(x) ≥ 0 bounded in the bounded domain $${\Omega \subset \mathbb R^N}$$. We prove that if $${N \ne 2m}$$ and $${\int_0^1 s^{-1} (\mathop{\rm meas}\nolimits \{x \in \Omega : |a(x)| \leq s \})^\theta {\rm d}s < \infty,\ \theta=\min\left(\frac{2m}N,1\right)}$$, then the solution u vanishes in a finite time. When N = 2m, the same property holds if $${\int_0^1 s^{-1} \left( \mathop{\rm meas}\nolimits \{x \in \Omega : |a(x)| \leq s \} \right) \ln \left( \mathop{\rm meas}\nolimits \{x \in \Omega : |a(x)| \leq s \} \right) {\rm d}s > - \infty}$$.

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