On higher-order semilinear parabolic equations with measures as initial data (Q1772561)

Summary: We consider \(2m\)th-order \((m\geq 2)\) semilinear parabolic equations \[ u_t=-(-\Delta)^m u\pm|u|^{p-1}u \quad\text{in }\mathbb R^N\times\mathbb R_+ \quad(p>1), \] with Dirac's mass \(\delta(x)\) as the initial function. We show that for \(p<p_0=1+2m/N\), the Cauchy problem admits a solution \(u(x,t)\) which is bounded and smooth for small \(t>0\), while for \(p\geq p_0\) such a local in time solution does not exist. This leads to a boundary layer phenomenon in constructing a proper solution via regular approximations.