| Summary: | We consider the Cauchy problem <inline-formula> <math display="inline"> <semantics> <mrow> <mrow> <mo>(</mo> <msub> <mi mathvariant="double-struck">D</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> </msub> <mi>u</mi> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <mi>λ</mi> <mi>u</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </semantics> </math> </inline-formula>, <inline-formula> <math display="inline"> <semantics> <mrow> <mi>u</mi> <mo>(</mo> <mn>0</mn> <mo>)</mo> <mo>=</mo> <mn>1</mn> </mrow> </semantics> </math> </inline-formula>, where <inline-formula> <math display="inline"> <semantics> <msub> <mi mathvariant="double-struck">D</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> </msub> </semantics> </math> </inline-formula> is the general convolutional derivative introduced in the paper (A. N. Kochubei, Integral Equations Oper. Theory <b>71</b> (2011), 583−600), <inline-formula> <math display="inline"> <semantics> <mrow> <mi>λ</mi> <mo>></mo> <mn>0</mn> </mrow> </semantics> </math> </inline-formula>. The solution is a generalization of the function <inline-formula> <math display="inline"> <semantics> <mrow> <mi>t</mi> <mo>↦</mo> <msub> <mi>E</mi> <mi>α</mi> </msub> <mrow> <mo>(</mo> <mi>λ</mi> <msup> <mi>t</mi> <mi>α</mi> </msup> <mo>)</mo> </mrow> </mrow> </semantics> </math> </inline-formula>, where <inline-formula> <math display="inline"> <semantics> <mrow> <mn>0</mn> <mo><</mo> <mi>α</mi> <mo><</mo> <mn>1</mn> </mrow> </semantics> </math> </inline-formula>, <inline-formula> <math display="inline"> <semantics> <msub> <mi>E</mi> <mi>α</mi> </msub> </semantics> </math> </inline-formula> is the Mittag−Leffler function. The asymptotics of this solution, as <inline-formula> <math display="inline"> <semantics> <mrow> <mi>t</mi> <mo>→</mo> <mo>∞</mo> </mrow> </semantics> </math> </inline-formula>, are studied.
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