Suppose that x'(t) + Ax(t) = f(t, x(t)), t ≥ 0 is a semilinear parabolic equation, e-At is bounded and f satisfies the usual continuity condition. If for some 0 < ≤ 1, 0 < < 1, p > 1, > 1 ||t Ae-At|| ≤ C, t ≥ 1 ||f(t, x)|| ≤ C(||A x|| p + (1 + t)-), t ≥ 0 whenever ||A x|| + ||x|| is small enough, then for small initial data there exist stable global solutions. Moreover, if the space is reflexive then their limit states exist. Some theorems that are useful for obtaining the above bounds and some examples are also presented.
Institute for Mathematics and Its Applications>IMA Preprints Series
Stability for Semilinear Parabolic Equations with Noninvertible Linear Operator.
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