## On a solution of the Cauchy problem in the weighted spaces of Beurling ultradistributions.(English)Zbl 1359.47037

Authors’ abstract: “Results of G. da Prato and E. Sinestrari [Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 14, No. 2, 285–344 (1987; Zbl 0652.34069)] on differential operators with non-dense domain but satisfying the Hille-Yosida condition, are applied in the setting of Beurling weighted spaces of ultradistributions $${\mathcal D}^{\prime(s)}_{L^p}((0,T)\times U)$$, where $$U$$ is open and bounded set in $$\mathbb{R}^d$$. For this purpose, new structural theorems are given for $${\mathcal D}^{\prime(s)}_{L^p}((0,T)\times U)$$. Then a class of Cauchy problems in the general setting of such spaces of ultradistributions is analyzed.”
More specifically, Da Prato and Sinestrari studied the Cauchy problem $u'(t)= Au(t)+ f(t)\,u(0)= u_0,\tag{1}$ where $$A$$ is a closed operator in a Banach space $$E$$, satisfying the Hille-Yosida conditions and not necessarily having a dense domain in $$E$$. The present paper extends the results of problem (1) to the distributional Schwartz spaces and Beurling ultradistributional spaces studied by the papers of H. Komatsu listed in the bibliography.
The paper continues to prove excellent results within the setting of Beurling and Roumieu spaces. I recommend a careful read to gather all the results contained in this lengthy, but clearly presented paper.

### MSC:

 47D03 Groups and semigroups of linear operators 34G10 Linear differential equations in abstract spaces 46F05 Topological linear spaces of test functions, distributions and ultradistributions

### Keywords:

Cauchy problem; strict weak solution; $$F$$-weak solution

Zbl 0652.34069
Full Text:

### References:

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