Abstract
The existence, uniqueness, regularity and asymptotic behavior of global solutions of semilinear heat equations in Hilbert spaces are studied by developing new results in the theory of one-parameter strongly continuous semigroups of bounded linear operators. Applications to special semilinear heat equations inL 2(ℝn)governed by pseudo-differential operators are given.
Highlights
Let X be a complex Hilbert space in which the norm and inner product are denoted by · and (·, ·) respectively
The following theorem is the basis for the existence, uniqueness and regularity of global solutions of semilinear heat equations in Hilbert spaces
A well-known corollary on the existence, uniqueness and regularity of global solutions of semilinear heat equations in Hilbert spaces is stated in Remark 3.3
Summary
Let X be a complex Hilbert space in which the norm and inner product are denoted by · and (·, ·) respectively. Semilinear heat equations, existence, uniqueness, regularity, asymptotic behavior. We can extend the linear operator A to the space S as follows: For any u in S , we define Au to be the element in S given by. Let A be a linear operator from X into X with domain S such that A maps S into S and its formal adjoint A∗ maps S into S continuously. The following theorem is the basis for the existence, uniqueness and regularity of global solutions of semilinear heat equations in Hilbert spaces. A well-known corollary on the existence, uniqueness and regularity of global solutions of semilinear heat equations in Hilbert spaces is stated in Remark 3.3. Theorems on the asymptotic stability of the equilibrium solutions and on the existence of absorbing sets for global solutions of semilinear heat equations are given in Sections 4 and 5. Semilinear heat equations modelled by specific ordinary and partial differential operators have been studied extensively in, e.g., [1, 2] by BellaniMorante and [7] by Tanabe
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