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Singular integral inequalities and stability of semilinear parabolic equations. (English) Zbl 0915.34057
The abstract Cauchy initial value problem given for $\tfrac {du}{dt} + Au = f(t,u), \quad u \in X, \qquad u(0)=u_0 \in X,$ is considered, where $$X$$ is an appropriate Banach space, $$A:X \to X$$ is a sectorial operator and $$f$$ has some more specific properties. Conditions are presented for the existence of a solution $$u$$ defined on $$[0,+\infty)$$ such that $$\lim _{t\to \infty }\| u(t)\| =0$$.

##### MSC:
 34G20 Nonlinear differential equations in abstract spaces 34D05 Asymptotic properties of solutions to ordinary differential equations 35K55 Nonlinear parabolic equations 35B35 Stability in context of PDEs 34G10 Linear differential equations in abstract spaces
##### Keywords:
integral inequality; parabolic equation; stability
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