The Super-Diffusive Singular Perturbation Problem
In this paper we study a class of singularly perturbed defined abstract Cauchy problems. We investigate the singular perturbation problem <inline-formula> <math display="inline"> <semantics> <mrow> <mrow>...
| Main Authors: | , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2020-03-01
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| Series: | Mathematics |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2227-7390/8/3/403 |
| Summary: | In this paper we study a class of singularly perturbed defined abstract Cauchy problems. We investigate the singular perturbation problem <inline-formula>
<math display="inline">
<semantics>
<mrow>
<mrow>
<mo>(</mo>
<msub>
<mi>P</mi>
<mi>ϵ</mi>
</msub>
<mo>)</mo>
</mrow>
<msup>
<mi>ϵ</mi>
<mi>α</mi>
</msup>
<msubsup>
<mi>D</mi>
<mi>t</mi>
<mi>α</mi>
</msubsup>
<msub>
<mi>u</mi>
<mi>ϵ</mi>
</msub>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
<mo>+</mo>
<msubsup>
<mi>u</mi>
<mi>ϵ</mi>
<mo>′</mo>
</msubsup>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mi>A</mi>
<msub>
<mi>u</mi>
<mi>ϵ</mi>
</msub>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
<mo>,</mo>
<mi>t</mi>
<mo>∈</mo>
<mrow>
<mo>[</mo>
<mn>0</mn>
<mo>,</mo>
<mi>T</mi>
<mo>]</mo>
</mrow>
</mrow>
</semantics>
</math>
</inline-formula>, <inline-formula>
<math display="inline">
<semantics>
<mrow>
<mn>1</mn>
<mo><</mo>
<mi>α</mi>
<mo><</mo>
<mn>2</mn>
<mo>,</mo>
<mi>ϵ</mi>
<mo>></mo>
<mn>0</mn>
<mo>,</mo>
</mrow>
</semantics>
</math>
</inline-formula> for the parabolic equation <inline-formula>
<math display="inline">
<semantics>
<mrow>
<mrow>
<mo>(</mo>
<mi>P</mi>
<mo>)</mo>
</mrow>
<msubsup>
<mi>u</mi>
<mn>0</mn>
<mo>′</mo>
</msubsup>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mi>A</mi>
<msub>
<mi>u</mi>
<mn>0</mn>
</msub>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
<mo>,</mo>
<mi>t</mi>
<mo>∈</mo>
<mrow>
<mo>[</mo>
<mn>0</mn>
<mo>,</mo>
<mi>T</mi>
<mo>]</mo>
</mrow>
<mo>,</mo>
</mrow>
</semantics>
</math>
</inline-formula> in a Banach space, as the singular parameter goes to zero. Under the assumption that <i>A</i> is the generator of a bounded analytic semigroup and under some regularity conditions we show that problem <inline-formula>
<math display="inline">
<semantics>
<mrow>
<mo>(</mo>
<msub>
<mi>P</mi>
<mi>ϵ</mi>
</msub>
<mo>)</mo>
</mrow>
</semantics>
</math>
</inline-formula> has a unique solution <inline-formula>
<math display="inline">
<semantics>
<mrow>
<msub>
<mi>u</mi>
<mi>ϵ</mi>
</msub>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
</mrow>
</semantics>
</math>
</inline-formula> for each small <inline-formula>
<math display="inline">
<semantics>
<mrow>
<mi>ϵ</mi>
<mo>></mo>
<mn>0</mn>
<mo>.</mo>
</mrow>
</semantics>
</math>
</inline-formula> Moreover <inline-formula>
<math display="inline">
<semantics>
<mrow>
<msub>
<mi>u</mi>
<mi>ϵ</mi>
</msub>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
</mrow>
</semantics>
</math>
</inline-formula> converges to <inline-formula>
<math display="inline">
<semantics>
<mrow>
<msub>
<mi>u</mi>
<mn>0</mn>
</msub>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
</mrow>
</semantics>
</math>
</inline-formula> as <inline-formula>
<math display="inline">
<semantics>
<mrow>
<mi>ϵ</mi>
<mo>→</mo>
<msup>
<mn>0</mn>
<mo>+</mo>
</msup>
<mo>,</mo>
</mrow>
</semantics>
</math>
</inline-formula> the unique solution of equation <inline-formula>
<math display="inline">
<semantics>
<mrow>
<mo>(</mo>
<mi>P</mi>
<mo>)</mo>
</mrow>
</semantics>
</math>
</inline-formula>. |
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| ISSN: | 2227-7390 |