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

Victor Galaktionov — 2004

Journal of the European Mathematical Society

We consider 2 m th-order ( m ≥ 2 ) semilinear parabolic equations u t = − ( − Δ ) m u ± | u | p − 1 u in ℛ N × ℝ + ( p > 1 ) , with Dirac’s mass δ ( x ) as the initial function. We show that for p < p 0 = 1 + 2 m / N , the Cauchy problem admits a solution u ( x , t ) which is bounded and smooth for small t > 0 , while for p ≥ 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.

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