Timeline for $u_t=Au+F(u)$ where $A$ is the infinitesimal generator of $C_0$-semigroup
Current License: CC BY-SA 4.0
13 events
| when toggle format | what | by | license | comment | |
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| Mar 30, 2020 at 15:58 | vote | accept | Math | ||
| Mar 29, 2020 at 21:33 | vote | accept | Math | ||
| Mar 30, 2020 at 15:58 | |||||
| Mar 29, 2020 at 15:40 | comment | added | Math | Just to clarify, does the proof of Proposition 4.3.9 depend on whether or not $F$ is locally Lipschitz? (It seems not) | |
| Mar 29, 2020 at 15:35 | comment | added | S. Maths | @VictorHugo Welcome! | |
| Mar 29, 2020 at 15:31 | vote | accept | Math | ||
| Mar 29, 2020 at 20:17 | |||||
| Mar 29, 2020 at 15:31 | comment | added | Math | Thank you! @S.Maths | |
| Mar 29, 2020 at 15:29 | comment | added | S. Maths | @VictorHugo see my edit | |
| Mar 29, 2020 at 15:27 | history | edited | S. Maths | CC BY-SA 4.0 |
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| Mar 29, 2020 at 12:46 | comment | added | Math | In Theorem $1.7$ you have the hypothesis that $f$ is Lipschitz continuous from $Y=D(A)$ into $Y$ with the graph norm (which is a strong condition.). After Theorem $1.7$ there is a remark saying that: If in the previous theorem we assume only that $f: [t_ 0, T] \times Y \rightarrow Y$ is locally Lipschitz continuous in $Y$ uniformly in $[t_0, T]$ we obtain, using Theorem 1.4, that for every $u_0 \in D(A) $ the initial value problem possesses a classical solution on a maximal interval $[t_0, t_\max[$. It would be interesting if this comment was valid with $f$ only locally Lipschitz in $X$. | |
| Mar 29, 2020 at 12:35 | comment | added | S. Maths | @VictorHugo for the second part see Theorem 1.7 (and theorems before) and for the third part see Theorem 1.2 (for global solution) as I said in my answer. | |
| Mar 29, 2020 at 11:55 | comment | added | Math | By using Theorem 1.4 I agree with the first part of Theorem 5.8, that is, If $F:H\rightarrow H$ is locally Lipschitz then for every $u \in H$, there is a unique mild solution $u:[0,T_{\max}[\rightarrow H$ of $(5.59)$ with either $T_{\max}=+\infty$, or $T_{\max}<\infty$ and $\lim_{t \uparrow T_{\max}}\|u(t)\|=+\infty$. I am still in doubt with respect to the second and third part of the theorem, I have not found the respective results in the Pazy's book. | |
| Mar 29, 2020 at 1:18 | history | edited | S. Maths | CC BY-SA 4.0 |
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| Mar 29, 2020 at 1:00 | history | answered | S. Maths | CC BY-SA 4.0 |