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On Coercive Solvability of Parabolic Equations with Variable Operators

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AbstractIn a Banach space E, the Cauchy problem υ′t+Atυt=ft0≤t≤1,υ0=υ0,$$ \upsilon^{\prime }(t)+A(t)\upsilon (t)=f(t)\kern1em \left(0\le t\le 1\right),\kern1em \upsilon (0)={\upsilon}_0, $$ is considered for a differential equation with linear strongly positive operator A(t)… Click to show full abstract

AbstractIn a Banach space E, the Cauchy problem υ′t+Atυt=ft0≤t≤1,υ0=υ0,$$ \upsilon^{\prime }(t)+A(t)\upsilon (t)=f(t)\kern1em \left(0\le t\le 1\right),\kern1em \upsilon (0)={\upsilon}_0, $$ is considered for a differential equation with linear strongly positive operator A(t) such that its domain D = D(A(t)) does not depend on t and is everywhere dense in E and A(t) generates an analytic semigroup exp{−sA(t)}(s ≥ 0). Under natural assumptions on A(t), we prove the coercive solvability of the Cauchy problem in the Banach space C0β,γ$$ {C}_0^{\beta, \upgamma} $$ (E). We prove a stronger estimate for the solution compared with estimates known earlier, using weaker restrictions on f(t) and v0.

Keywords: variable operators; coercive solvability; equations variable; solvability; parabolic equations; solvability parabolic

Journal Title: Journal of Mathematical Sciences
Year Published: 2019

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