Semismall perturbations, semi-intrinsic ultracontractivity, and integral representations of nonnegative solutions for parabolic equations

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We consider nonnegative solutions of a parabolic equation in a cylinder D×I, where D is a noncompactdomain of a Riemannian manifold and I = (0,T ) with 0<T ∞or I = (−∞, 0). Under the assumption[SSP] (i.e., the constant function 1 is a semismall perturbation of the associated elliptic operator on D),we establish an integral representation theorem of nonnegative solutions: In the case I = (0,T ), any nonnegativesolution is represented uniquely by an integral on (D × {0}) ∪ (∂MD × [0,T )), where ∂MD isthe Martin boundary of D for the elliptic operator; and in the case I = (−∞, 0), any nonnegative solutionis represented uniquely by the sum of an integral on ∂MD × (−∞, 0) and a constant multiple of aparticular solution. We also show that [SSP] implies the condition [SIU] (i.e., the associated heat kernel issemi-intrinsically ultracontractive).


  • Journal of Functional Analysis

    Journal of Functional Analysis (257), 2009-07

    Elsevier Inc.


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