Equations of parabolic type in a Banach space

Sobolevskii, P. E.

doi:10.1090/trans2/049/01

Cited by 138 publications

(143 citation statements)

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“…We use the integral equation approach developed by Sobolevskii [23], and Fujita and Kato [9]. In § 4 we give applications to semilinear parabolic equations.…”

Section: These Conditions Also Imply S(t) Maps CL (D(a)) Into D(a) Fomentioning

confidence: 99%

“…where T is a linear operator in a complex Banach space X and F is a function with domain and range in X. Equations of this type have been studied by Sobolevskii [23], Fujita and Kato [9], Friedman [8], Henry [10] and others. We establish analyticity in t of solutions u{t) of this equation under suitable conditions on T and F. In particular, we assume that…”

Section: Ee^av(t)mentioning

confidence: 99%

“…The differential equation du/dt + Au = Fu is transformed into the integral equation (3.7) below. This method was introduced by Sobolevskii [23] and Fujita and Kato [9] and is now standard. We use methods similar to Henry [10], and therefore we are as brief as possible.…”

Section: U{t) = U(t)u(0) + γ U(t-s)f(s)dsmentioning

confidence: 99%

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Nonlinear holomorphic semigroups

Hayden¹,

Massey²

1975

Pacific J. Math.

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“…We use the integral equation approach developed by Sobolevskii [23], and Fujita and Kato [9]. In § 4 we give applications to semilinear parabolic equations.…”

Section: These Conditions Also Imply S(t) Maps CL (D(a)) Into D(a) Fomentioning

confidence: 99%

Section: Ee^av(t)mentioning

confidence: 99%

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Nonlinear holomorphic semigroups

Hayden¹,

Massey²

1975

Pacific J. Math.

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“…On the other hand, when D(A(t)) is constant, Tanabe [12] proved that the solution of (0.2) is twice differentiate if A(t)A(s)~ι is Holder continuously differentiate. P. E. Sobolevskii [9] showed that if…”

Section: From the Above It Is Clear That If Further Ait)' 1 E C°°[0 mentioning

confidence: 99%

The higher order differentiability of solutions of abstract evolution equations

Suryanarayana¹

1967

Pacific J. Math.

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“…1) in which A and {Bj} depend on time t, we can make use of the results in [11], [14] and [15]. However Arima succeeded in constructing Green function for the parabolic boundary problem and obtained the estimates on it in her…”

Section: Remarkmentioning

confidence: 99%

On semi-linear parabolic partial differential equations

Asano¹

1965

Publ. Res. Inst. Math. Sci.

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By Kiyoshi ASANOWe give an operator theoretical treatment of initial value problems for semi-linear parabolic partial differential equations : existence of solutions, uniqueness and regularity. We make use of the theory of fractional powers of operators, the theory of semigroups of operators and L^-estimates for elliptic boundary value problems. § 0. Introduction The purpose of this paper is to derive some results on the initial value problems for semi-linear parabolic partial differential equations. The equation is as follows :Here G is a bounded domain in Euclidean ^-space E n , A is an elliptic partial differential operator on G of order 2m and {Bj}^ is a system of m differential operators on the boundary of G. A and {Bj} satisfies some algebraic conditions ((J?) and (C) in § 3). F may contain the derivatives D%u of u of order less than 2m in its variables.If

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Equations of parabolic type in a Banach space

Cited by 138 publications

References 0 publications

Nonlinear holomorphic semigroups

Nonlinear holomorphic semigroups

The higher order differentiability of solutions of abstract evolution equations

On semi-linear parabolic partial differential equations

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