Доведено коректну розв'язність багатоточкової за часом задачі для еволюційних рівнянь з функціями від оператора диференціювання, зокрема, з операторами дробового диференціювання, та початковою функцією, яка є елементом простору узагальнених функцій типу $S'$. Досліджено поведінку розв'язку зазначеної задачі при $t \to +\infty$ у просторах типу $S'$ (слабка стабілізація). Знайдено умову на початкову функцію, при виконанні якої розв'язок стабілізується до нуля рівномірно на $\mathbb{R}$.

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