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> <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>&lt;</mo> <mi>α</mi> <mo>&lt;</mo> <mn>2</mn> <mo>,</mo> <mi>ϵ</mi> <mo>&gt;</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>&gt;</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>.

Abstract licence: CC BY

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