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**On well-posedness of impulsive problems for nonlinear parabolic equations.**
*(English)*
Zbl 1006.35055

The paper deals with the well-posedness of the Cauchy problem for the impulsive system governed by the abstract evolution equation with sectorial operator in Banach space and impulses at fixed times. This research has been motivated by the recent papers by the reviewer [Dyn. Contin. Discrete Impulsive Syst. 3, 57-88 (1997; Zbl 0879.34014)] where this and many related problems have been discussed, and by J. H. Liu [ibid. 6, 77-85 (1999; Zbl 0932.34067)], where a similar problem has been studied for the impulsive system with general, not necessarily sectorial, operator, generating a strongly continuous semigroup. The authors prove results on existence, uniqueness, and continuous dependence of solutions of the impulsive evolution system analogous to those established by the reviewer and Liu, although under slightly modified assumptions and by using a slightly different technique.

There are several serious omissions in the paper under review. First, a very important assumption of Hölder continuity in \(t\) of the function \(f\) required for the existence of solutions of abstract parabolic equations is omitted. Second, assumption of Theorem 1 (existence and uniqueness) \[ \frac{KC}{1-\alpha }\max \left[ e^{-a\gamma },1\right] \left( \gamma ^{1-\alpha }+\left|a\right|\gamma ^{2-\alpha }\right) <1 \tag{3} \] requires that either the Lipschitz constant \(K\) for the function \(f\), or the distance between the impulse instants \(\gamma =\max_{1\leq i\leq p+1}\left|t_{i}-t_{i-1}\right|\) is small. In the former case, this assumption does not differ much from those by the reviewer and Liu, while in the latter case too many impulses might be required to guarantee local existence of solutions. Since \(\alpha \) in assumption (H2) satisfies \(0\leq \alpha \leq 1,\) assumption (3) simply fails to hold if \(\alpha =1,\) and is difficult to satisfy if \(\alpha \) is close to \(1.\) Thus, existence of solutions in this case cannot be established. Finally, the authors assume only continuity of impulse operators, and this unusually mild condition is based on the speculation that “an impulsive problem is expected to be well-posed on a large interval of time even if its corresponding Cauchy problem blows up in a finite period.” This, in general, is false, and there are numerous examples in the literature showing that well-posed problems for differential equations start behaving “wildly” after the impulses are introduced (splitting and gluing of solutions, beating phenomenon, etc.), although by choosing impulsive operators intelligently, one can really improve the behavior of solutions. The latter, however, requires additional assumptions on impulse operators, and just continuity might not be enough.

There are several serious omissions in the paper under review. First, a very important assumption of Hölder continuity in \(t\) of the function \(f\) required for the existence of solutions of abstract parabolic equations is omitted. Second, assumption of Theorem 1 (existence and uniqueness) \[ \frac{KC}{1-\alpha }\max \left[ e^{-a\gamma },1\right] \left( \gamma ^{1-\alpha }+\left|a\right|\gamma ^{2-\alpha }\right) <1 \tag{3} \] requires that either the Lipschitz constant \(K\) for the function \(f\), or the distance between the impulse instants \(\gamma =\max_{1\leq i\leq p+1}\left|t_{i}-t_{i-1}\right|\) is small. In the former case, this assumption does not differ much from those by the reviewer and Liu, while in the latter case too many impulses might be required to guarantee local existence of solutions. Since \(\alpha \) in assumption (H2) satisfies \(0\leq \alpha \leq 1,\) assumption (3) simply fails to hold if \(\alpha =1,\) and is difficult to satisfy if \(\alpha \) is close to \(1.\) Thus, existence of solutions in this case cannot be established. Finally, the authors assume only continuity of impulse operators, and this unusually mild condition is based on the speculation that “an impulsive problem is expected to be well-posed on a large interval of time even if its corresponding Cauchy problem blows up in a finite period.” This, in general, is false, and there are numerous examples in the literature showing that well-posed problems for differential equations start behaving “wildly” after the impulses are introduced (splitting and gluing of solutions, beating phenomenon, etc.), although by choosing impulsive operators intelligently, one can really improve the behavior of solutions. The latter, however, requires additional assumptions on impulse operators, and just continuity might not be enough.

Reviewer: Yuri V.Rogovchenko (Famagusta)