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

• Volume: 006, Issue: 2, page 193-205
• ISSN: 1435-9855

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## Abstract

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We consider $2m$th-order $\left(m\ge 2\right)$ semilinear parabolic equations ${u}_{t}=-{\left(-\Delta \right)}^{m}u±{|u|}^{p-1}u$ in ${ℛ}^{N}×{ℝ}_{+}$$\left(p>1\right)$, with Dirac’s mass $\delta \left(x\right)$ as the initial function. We show that for $p<{p}_{0}=1+2m/N$, the Cauchy problem admits a solution $u\left(x,t\right)$ which is bounded and smooth for small $t>0$, while for $p\ge {p}_{0}$ such a local in time solution does not exist. This leads to a boundary layer phenomenon in constructing a proper solution via regular approximations.

## How to cite

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Galaktionov, Victor. "On higher-order semilinear parabolic equations with measures as initial data." Journal of the European Mathematical Society 006.2 (2004): 193-205. <http://eudml.org/doc/277209>.

@article{Galaktionov2004,
abstract = {We consider $2m$th-order $(m\ge 2)$ semilinear parabolic equations $u_t=−(−\Delta )^mu\pm |u|^\{p−1\}u$ in $\mathcal \{R\}^N\times \mathbb \{R\}_+$$(p>1), with Dirac’s mass \delta (x) as the initial function. We show that for p<p_0=1+2m/N, the Cauchy problem admits a solution u(x,t) which is bounded and smooth for small t>0, while for p\ge p_0 such a local in time solution does not exist. This leads to a boundary layer phenomenon in constructing a proper solution via regular approximations.}, author = {Galaktionov, Victor}, journal = {Journal of the European Mathematical Society}, keywords = {semilinear parabolic equations; Cauchy problem; boundary layer phenomenon; regular approximations; Dirac mass; existence; nonexistence; boundary layer phenomenon; regular approximations; Dirac mass; existence; nonexistence}, language = {eng}, number = {2}, pages = {193-205}, publisher = {European Mathematical Society Publishing House}, title = {On higher-order semilinear parabolic equations with measures as initial data}, url = {http://eudml.org/doc/277209}, volume = {006}, year = {2004}, } TY - JOUR AU - Galaktionov, Victor TI - On higher-order semilinear parabolic equations with measures as initial data JO - Journal of the European Mathematical Society PY - 2004 PB - European Mathematical Society Publishing House VL - 006 IS - 2 SP - 193 EP - 205 AB - We consider 2mth-order (m\ge 2) semilinear parabolic equations u_t=−(−\Delta )^mu\pm |u|^{p−1}u in \mathcal {R}^N\times \mathbb {R}_+$$(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 $p\ge p_0$ such a local in time solution does not exist. This leads to a boundary layer phenomenon in constructing a proper solution via regular approximations.
LA - eng
KW - semilinear parabolic equations; Cauchy problem; boundary layer phenomenon; regular approximations; Dirac mass; existence; nonexistence; boundary layer phenomenon; regular approximations; Dirac mass; existence; nonexistence
UR - http://eudml.org/doc/277209
ER -

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