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In this article, we study the blow-up of the damped wave equation in the \textit{scale-invariant case} and in the presence of two nonlinearities. More precisely, we consider the following equation: $$u_{tt}-\Delta u+\frac{\mu}{1+t}u_t=|u_t|^p+|u|^q, \quad \mbox{in}\ \R^N\times[0,\infty), $$ with small initial data.\\ For $\mu < \frac{N(q-1)}{2}$ and $\mu \in (0, \mu_*)$, where $\mu_*>0$ is depending on the nonlinearties' powers and the space dimension ($\mu_*$ satisfies $(q-1)\left((N+2\mu_*-1)p-2\right) = 4$), we prove that the wave equation, in this case, behaves like the one without dissipation ($\mu =0$). Our result completes the previous studies in the case where the dissipation is given by $\frac{\mu}{(1+t)^\beta}u_t; \ \beta >1$ (\cite{LT3}), where, contrary to what we obtain in the present work, the effect of the damping is not significant in the dynamics. Interestingly, in our case, the influence of the damping term $\frac{\mu}{1+t}u_t$ is important.

In this article, we investigate the blow-up for local solutions to a semilinear wave equation in the generalized Einstein - de Sitter spacetime with nonlinearity of derivative type. More precisely, we consider a semilinear damped wave equation with a time-dependent and not summable speed of propagation and with a time-dependent coefficient for the linear damping term with critical decay rate. We prove in this work that the results obtained in a previous work, where the damping coefficient takes two particular values $0$ or $2$, can be extended for any positive damping coefficient. In the blow-up case, the upper bound of the exponent of the nonlinear term is given, and the lifespan estimate of the global existence time is derived as well.

We consider the semilinear heat equation $$\partial_t u -\Delta u =f(u), \quad (x,t)\in \mathbb{R}^N\times [0,T),\qquad (1)$$ with $f(u)=|u|^{p-1}u\log^a (2+u^2)$, where $p>1$ is Sobolev subcritical and $a\in \mathbb{R}$. We first show an upper bound for any blow-up solution of (1). Then, using this estimate and the logarithmic property, we prove that the exact blow-up rate of any singular solution of (1) is given by the ODE solution associated with (1), namely $u' =|u|^{p-1}u\log^a (2+u^2)$. In other terms, all blow-up solutions in the Sobolev subcritical range are Type I solutions. Up to our knowledge, this is the first determination of the blow-up rate for a semilinear heat equation where the main nonlinear term is not homogeneous., Comment: arXiv admin note: text overlap with arXiv:2012.05374

We consider the semilinear wave equation $$\partial_t^2 u -\Delta u =f(u), \quad (x,t)\in \mathbb R^N\times [0,T),\qquad (1)$$ with $f(u)=|u|^{p-1}u\log^a (2+u^2)$, where $p>1$ and $a\in \mathbb R$, with subconformal power nonlinearity. We will show that the blow-up rate of any singular solution of (1) is given by the ODE solution associated with $(1)$, The result in one space dimension, has been proved in \cite{HZjmaa2020}. Our goal here is to extend this result to higher dimensions., Comment: arXiv admin note: substantial text overlap with arXiv:1906.12059

In this paper, we establish blow-up results for the semilinear wave equation in generalized Einstein-de Sitter spacetime with nonlinearity of derivative type. Our approach is based on the integral representation formula for the solution to the corresponding linear problem in the one-dimensional case, that we will determine through Yagdjian's Integral Transform approach. As upper bound for the exponent of the nonlinear term, we discover a Glassey-type exponent which depends both on the space dimension and on the Lorentzian metric in the generalized Einstein-de Sitter spacetime.

In this article, we consider the damped wave equation in the \textit{scale-invariant case} with time-dependent speed of propagation, mass term and time derivative nonlinearity. More precisely, we study the blow-up of the solutions to the following equation: $$ (E) \quad u_{tt}-t^{2m}\Delta u+\frac{\mu}{t}u_t+\frac{\nu^2}{t^2}u=|u_t|^p, \quad \mbox{in}\ \mathbb{R}^N\times[1,\infty), $$ that we associate with small initial data. Assuming some assumptions on the mass and damping coefficients, $\nu$ and $\mu>0$, respectively, that the blow-up region and the lifespan bound of the solution of $(E)$ remain the same as the ones obtained for the case without mass, {\it i.e.} $\nu=0$ in $(E)$. The latter case constitutes, in fact, a shift of the dimension $N$ by $\frac{\mu}{1+m}$ compared to the problem without damping and mass. Finally, we think that the new bound for $p$ is a serious candidate to the critical exponent which characterizes the threshold between the blow-up and the global existence regions.

We study in this article the blow-up of the solution of the generalized Tricomi equation in the presence of two mixed nonlinearities, namely we consider $$ (Tr) \hspace{1cm} u_{tt}-t^{2m}\Delta u=|u_t|^p+|u|^q, \quad \mbox{in}\ \mathbb{R}^N\times[0,\infty),$$ with small initial data, where $m\ge0$.\\ For the problem $(Tr)$ with $m=0$, which corresponds to the uniform wave speed of propagation, it is known that the presence of mixed nonlinearities generates a new blow-up region in comparison with the case of a one nonlinearity ($|u_t|^p$ or $|u|^q$). We show in the present work that the competition between the two nonlinearities still yields a new blow region for the Tricomi equation $(Tr)$ with $m\ge0$, and we derive an estimate of the lifespan in terms of the Tricomi parameter $m$. As an application of the method developed for the study of the equation $(Tr)$ we obtain with a different approach the same blow-up result as in \cite{Lai2020} when we consider only one time-derivative nonlinearity, namely we keep only $|u_t|^p$ in the right-hand side of $(Tr)$., Comment: 17 pages. arXiv admin note: text overlap with arXiv:2008.02109

We consider in this article the damped wave equation, in the \textit{scale-invariant case} with combined two nonlinearities, which reads as follows: \begin{displaymath} \d (E) \hspace{1cm} u_{tt}-\Delta u+\frac{\mu}{1+t}u_t=|u_t|^p+|u|^q, \quad \mbox{in}\ \R^N\times[0,\infty), \end{displaymath} with small initial data.\\ Compared to our previous work \cite{Our}, we show in this article that the first hypothesis on the damping coefficient $\mu$, namely $\mu < \frac{N(q-1)}{2}$, can be removed, and the second one can be extended from $(0, \mu_*/2)$ to $(0, \mu_*)$ where $\mu_*>0$ is solution of $(q-1)\left((N+\mu_*-1)p-2\right) = 4$. Indeed, owing to a better understanding of the influence of the damping term in the global dynamics of the solution, we think that this new interval for $\mu$ describe better the threshold between the blow-up and the global existence regions. Moreover, taking advantage of the techniques employed in the problem $(E)$, we also improve the result in \cite{LT2,Palmieri} in relationship with the Glassey conjecture for the solution of $(E)$ but without the nonlinear term $|u|^q$. More precisely, we extend the blow-up region from $p \in (1, p_G(N+\sigma)]$, where $\sigma$ is given by \eqref{sigma} below, to $p \in (1, p_G(N+\mu)]$ giving thus a better estimate of the lifespan in this case., Comment: arXiv admin note: text overlap with arXiv:2006.12600

We construct a finite-time blow-up solution for a class of strongly perturbed semilinear wave equation with an isolated characteristic point in one space dimension. Given any integer k ≥ 2 and ζ 0 ∈ R , we construct a blow-up solution with a characteristic point a, such that the asymptotic behavior of the solution near ( a , T ( a ) ) shows a decoupled sum of k solitons with alternate signs, whose centers (in the hyperbolic geometry) have ζ 0 as a center of mass, for all times. Although the result is similar to the unperturbed case in its statement, our method is new. Indeed, our perturbed equation is not invariant under the Lorentz transform, and this requires new ideas. In fact, the main difficulty in this paper is to prescribe the center of mass ζ 0 ∈ R . We would like to mention that our method is valid also in the unperturbed case, and simplifies the original proof by Cote and Zaag [9] , as far as the center of mass prescription is concerned.

We consider the semilinear wave equation $$\partial_t^2 u -\Delta u =f(u), \quad (x,t)\in \mathbb{R}^N\times [0,T),\qquad (1)$$ with $f(u)=|u|^{p-1}u\log^a (2+u^2)$, where $p>1$ and $a\in \mathbb{R}$. We show an upper bound for any blow-up solution of (1). Then, in the one space dimensional case, using this estimate and the logarithmic property, we prove that the exact blow-up rate of any singular solution of (1) is given by the ODE solution associated with $(1)$, namely $u'' =|u|^{p-1}u\log^a (2+u^2)$ Unlike the pure power case ($g(u)=|u|^{p-1}u$) the difficulties here are due to the fact that equation (1) is not scale invariant., Comment: 37 pages

We study a class of perturbations for the semilinear wave equation with critical power nonlinearity (in the conformal transform sense). Working in the framework of similarity variables, we introduce a Lyapunov functional for this problem. Using a two-step argument based on interpolation and a critical Gagliardo–Nirenberg inequality, we establish that the blow-up rate of any singular solution is given by the solution of the nonperturbed associated ODE, specifically u″ = up.

We consider the damped hyperbolic equation (1) e u τ τ + u τ = ( a ( ξ ) u ξ ) ξ − | u | p − 1 u , ( ξ , τ ) ∈ R × R + , where e > 0 , a ( ξ ) → 1 as | ξ | → + ∞ and 1 p 3 . We prove in this article that the exact self-similar solutions of the semi-linear parabolic equation obtained by setting e = 0 and a ( ξ ) ≡ 1 in (1) are also asymptotically stable self-similar solutions of the Eq. (1) . The proof of our result relies on various energy estimates rewritten in the variables ξ / τ , ln τ .

In this paper, we study a perturbation of the Navier–Stokes equations which is given by a particular case of a third-grade fluid in . We prove that the Cauchy problem is globally well posed in the energy space . We also prove a regularity propagation result: if the initial data are in , the solution belongs to for all time and we give a control on this norm.

We consider in this work some class of strongly perturbed for the semilinear wave equation with conformal power nonlinearity. We obtain an optimal estimate for a radial blow-up solution and we have also obtained two less stronger estimates. These results are achieved in three-steps argument by the construction of a Lyapunov functional in similarity variables and the Pohozaev identity derived by multiplying (1.14) by y∂yw.

We consider in this paper a large class of perturbed semilinear wave equations with critical (in the conformal transform sense) power nonlinearity. We will show that the blow-up rate of any singular solution is given by the solution of the non-perturbed associated ODE. The result in the radial case has been proved in [13]. The same approach will be followed here, but the main difference is to construct a Lyapunov functional in similarity variables valid in the non-radial case, which is far from being trivial. That functional is obtained by combining some classical estimates and a new identity of the Pohozaev type., Comment: 25 pages

Glioblastomas are the most common primary brain tumor, but despite aggressive treatment, continue to be marked by a poor prognosis, with a median survival of 14.6 months for glioblastoma (GBM).1,2 These tumors at recurrence are particularly challenging given the lack of effective salvage therapy, with 6 month progression free survival (PFS-6) ranging from 3–15% in patients with GBM.3,4 Angiogenesis in glioblastomas is fueled by vascular endothelial growth factor (VEGF) signaling, and blockade of this pathway is associated with rapid radiographic response as shown by investigations involving bevacizumab, a humanized monoclonal antibody that targets vascular endothelial growth factor A (VEGF-A), and cediranib, a pan-VEGF receptor tyrosine kinase inhibitor.5–7 Bevacizumab has received accelerated regulatory approval for use as salvage chemotherapy against recurrent GBM in the US and appears to prolong progression free survival and reducing symptom burden, though improvement in overall survival has not been effectively demonstrated.5,7 However, bevacizumab has not universally received approval (for instance, in Europe) for use in recurrent glioblastoma due to the absence of survival benefit from a randomized controlled trial.8 Recently completed phase III clinical trials using bevacizumab in newly diagnosed glioblastoma patients have also shown improvement in PFS but failed to reveal a significant benefit to overall survival.9,10 Glioblastomas that progress during bevacizumab therapy tend to be unresponsive to additional therapy.11–13 Even with a second bevacizumab containing regimen, response is poor with retrospective studies showing a PFS-6 of

The survival outcome of patients with malignant gliomas is still poor, despite advances in surgical techniques, radiation therapy and the development of novel chemotherapeutic agents. The heterogeneity of molecular alterations in signaling pathways involved in the pathogenesis of these tumors contributes significantly to their resistance to treatment. Several molecular targets for therapy have been discovered over the last several years. Therapeutic agents targeting these signaling pathways may provide more effective treatments and may improve survival. This review summarizes the important molecular therapeutic targets and the outcome of published clinical trials involving targeted therapeutic agents in glioma patients.

We give blow-up results for the Klein–Gordon equation and other perturbations of the semilinear wave equations with superlinear power nonlinearity, in one space dimension or in higher dimension under radial symmetry outside the origin.

Video encryption is an important field. It has been recently rapidly used in many multimedia systems. The proposed method is focusing on H.264/AVC [1] [2] video coding standard. Several encryption algorithms are based on intra frame/slice had been utilized. The drawbacks of previously used algorithms is that they are vulnerable to known plaintext attack, the increase of stream size, they can't handle real-time processing and they face bandwidth limitations of bit rate [3]. The proposed algorithm is to secure a video during encoding process. The proposed method uses Selective Encryption(SE) [4] to encrypt a CAVLC (Context Adaptive Variable Length Code) stage in order to minimize the amount of bits on compressed file. The security is based on the security of stream cipher. The most important factor here is the compact ratio, Bit rate and Frame rate.

We consider in this paper a large class of perturbed semilinear wave equation with subconformal power nonlinearity. In particular, we allow log perturbations of the main source. We derive a Lyapunov functional in similarity variables and use it to derive the blow-up rate. Throughout this work, we use some techniques developed for the unperturbed case studied by Merle and Zaag [12] together with ideas introduced by Hamza and Zaag in [5] for a class of weather perturbations., 17 pages. arXiv admin note: substantial text overlap with arXiv:1002.2328

We consider the semilinear wave equation in higher dimensions with power nonlinearity in the superconformal range, and its perturbations with lower order terms, including the Klein-Gordon equation. We improve the upper bounds on blow-up solutions previously obtained by Killip, Stovall and Visan [22]. Our proof uses the similarity variables' setting. We consider the equation in that setting as a perturbation of the conformal case, and we handle the extra terms thanks to the ideas we already developed in [16] for perturbations of the pure power conformal case with lower order terms.

Bevacizumab (BEV) is widely used for treatment of patients with recurrent glioblastoma. It is not known if there are differences in outcome between early versus delayed BEV treatment of recurrent glioblastoma. We examined the relationship between the time of starting BEV treatment and outcomes in patients with recurrent glioblastoma. In this retrospective chart review, we identified patients with recurrent glioblastoma diagnosed between 2005 and 2011 who were treated with BEV alone or BEV-containing regimens. Data was analyzed to determine overall survival (OS) from time of diagnosis and progression free survival (PFS) from time of starting BEV. A total of 298 patients were identified, 112 patients received early BEV, 133 patients received delayed BEV, and 53 patients were excluded because they either progressed within 3 months of radiation or received BEV at the time of diagnosis. There was no significant difference in PFS between patients that received early BEV and those that received delayed BEV (5.2 vs. 4.3 months, p = 0.2). Patients treated with delayed BEV had longer OS when compared to those treated with early BEV (25.9 vs. 20.8 months, p = 0.005). In patients with recurrent glioblastoma, there was no significant difference in PFS from the time of starting BEV between early and delayed BEV. Although patients treated with delayed BEV seemed to have longer OS, a conclusion regarding OS outcome requires further prospective trials. These results may indicate that delaying treatment with BEV is not detrimental for survival of patients with recurrent glioblastoma.

2075 Background: Bevacizumab (bev) is widely used for treatment of patients with recurrent glioblastoma (GB). Adverse effects (AEs) to bev or bev-containing regimens can cause discontinuation of tr...

2042 Background: Bevacizumab (BEV) is widely used for treatment of patients with recurrent glioblastoma (GB). Differences in outcome between early versus delayed BEV treatment of recurrent GB are not well defined. We examined the relationship between the time of start of BEV treatment and outcomes in patients with recurrent GB. Methods: In this retrospective chart review derived from our longitudinal database, we identified patients with recurrent GB between 2001 and 2011, who were treated with BEV alone or BEV-containing regimens. Data was analyzed to determine overall survival (OS) from time of diagnosis and progression free survival (PFS) from time of BEV start. Early BEV was defined as start of BEV treatment at first recurrence, while delayed BEV was defined as start of treatment at second recurrence or later. Results: A total of 298 patients with recurrent GB who received BEV were identified, of whom 149 patients received early BEV, 134 patients received delayed BEV, and 15 patients who were excluded because they received BEV upfront. There were no significant differences in the age, sex, performance status and extent of resection between patients treated with early BEV and those treated with delayed BEV. The median time from diagnosis to first recurrence was more than 6 months (mos.) for both groups (6.5 mos. for early BEV and 7.6 mos. for delayed BEV, p = 0.01). The median time from diagnosis to start of BEV was 7.9 mos. for patients with early BEV and 15.6 mos. for patients with delayed BEV (p

2064 Background: Bev is the standard treatment for patients with recurrent glioblastoma (GB) but is also used in treating recurrent anaplastic gliomas (AG). Differences in outcome between these groups and optimal duration of treatment with Bev in pts with recurrent malignant gliomas are not well defined. We examined the relationship between the duration of Bev treatment and the outcome in pts with GB and AG. Methods: In this retrospective chart and data review derived from our longitudinal database, we identified pts with recurrent AG and GB who were treated with Bev alone or Bev-containing regimens between 2005 and 2009; the data was analyzed to determine the overall survival (OS) and the progression free survival (PFS). Results: A total of 261 patients with recurrent malignant gliomas (196 with GB and 65 with AG) were identified. There was no significant difference between the median length of treatment between AG and GB (5.81±0.66 months vs. 6.77±0.52 months, p=0.32). PFS6 was 34.2% (95% CI, 27.8-41.3) for patients with GB and 44.2% (95% CI, 32.5-56.7) for patients with AG. Patients with GB who were treated ≥6 months had a significantly higher OS (29.13 months vs. 20.16 months, p= 0.001) compared to those treated

2030 Background: BEV is approved for use in recurrent glioblastoma. Patients (pts) who benefit from BEV therapy are often treated until tumor progression but fail to respond to salvage therapy suggesting that BEV may alter tumor biology. In a subset of pts who benefit from BEV, treatment is discontinued for reasons other than disease progression; the characteristics and outcomes of this subset are poorly defined. Methods: In this IRB approved retrospective study, our neuro-oncology longitudinal database was screened for pts treated with BEV for ≥ 6 months (mo) between 2005-2010 and 18 pts were identified in whom BEV was discontinued for reasons other than disease progression (BEV-D group). A cohort of 72 pts who received BEV until treatment failure due to progression was used as comparator (BEV-F group). Results: In the BEV-D group, 5 pts completed a planned treatment course and 13 stopped BEV due to toxicity; in this group, progression free survival at 12 mo (PFS12) was 83.3% (95% CI, 56.8-94.3) and median time to progression (TTP) 27.6 mo. Median TTP after BEV discontinuation was 7.0 mo. In contrast, in the BEV-F group, PFS12 was 24.6% (95% CI, 13.9-36.2) and median PFS 9.7 mo. Length of BEV therapy was not significantly different between the groups with a median time to discontinuation of 10.2 and 12 mo. In 12/18 pts in the BEV-D group who subsequently had tumor recurrence a predominantly local pattern of progression was seen unlike those in the BEV-F group who had more infiltrative or distant failures. Salvage therapy yielded a PFS-6 of 28.6% (95% CI, 4.1-61.2) with a median PFS of 17.1 wk compared with 6.8% (95% CI, 1.2-19.8) and 9 wk in the comparator group. Conclusions: Among pts who benefit from BEV therapy, the BEV-D group did not experience an immediate progression suggesting continued benefit after BEV cessation. This cohort also had a less invasive pattern of recurrence and a possibly improved response to salvage therapy compared with BEV-F group. Our results suggest that planned cessation of BEV therapy could potentially change patterns of progression and response to subsequent therapy. This strategy warrants further evaluation in prospective studies given the absence of effective salvage therapy after BEV failure.