Sentence examples similar to obtained blow from inspiring English sources

Under certain conditions, they obtained blow-up criteria.

Korpusov [30] considered a Sobolev type equation with a nonlocal source and obtained blow-up results under suitable conditions on initial data and nonlinear function.

Caflisch et al. [28] and Zhang and Liu [29] obtained blow-up criteria of smooth solutions to 3-D ideal MHD equations, respectively.

For the case (D(U =U^{n}), (ngeq0), Kowalczyk [2] obtained blow-up control relies on the presence of a pressure function, which increases faster than a logarithm for high enough cells densities: for such a pressure function the solutions cannot blow-up in a finite time, and proved the global boundedness for (1.1) under some other conditions.

In this paper, we prove the local well-posedness for the incompressible porous media equation in Triebel-Lizorkin spaces and obtain blow-up criterion of smooth solutions.

We note that we obtain blow-up, even if the coefficients are constants and also for K i ( t ) functions decreasing not too fast at infinity.

Remark 2.1 The maximal time of existence T > 0 in Theorem 2.1 is independent of the Sobolev index s > 3 / 2. Next we present the Sobolev-type inequalities, which play a key role to obtain blow-up results for the Cauchy problem (1.1) in the sequel.

Moreover, they also obtained the blow up of weak solutions.

In particular, Kong and Wang in [29], by using some ideas of Souplet [35], obtained the blow-up conditions and blow-up profile of the following system: (1.10).

and showed that blow-up occurs at some finite time under certain conditions on the nonlinearities and the data, upper and lower bounds for the blow-up time were obtained when blow-up occurs.

They obtained the blow-up condition and its blow-up rate estimate.