Decay properties of solutions to the Cauchy problem for the damped wave equation with absorption

AbstractWe consider the Cauchy problem for the damped wave equation with absorptionutt−Δu+ut+|u|p−1u=0,(t,x)∈R+×RN. The behavior of u as t→∞ is expected to be the same as that for the corresponding heat equationϕt−Δϕ+|ϕ|p−1ϕ=0,(t,x)∈R+×RN, which has the similarity solution wa(t,x) with the form t−1/(p−1)f(x/t) depending on a=lim|x|→∞|x|2/(p−1)f(x)⩾0 provided that p is less than the Fujita exponent pc(N):=1+2/N. In this paper, as a first step, if 1<p<pc(N) and the data (u0,u1)(x) decays exponentially as |x|→∞ without smallness condition, the solution is shown to decay with rates as t→∞,(∗)(‖u(t)‖L2,‖u(t)‖Lp+1,‖∇u(t)‖L2)=O(t−1p−1+N4,t−1p−1+N2(p+1),t−1p−1−12+N4), those of which seem to be reasonable, because the similarity solution wa(t,x) has the same decay rates as (∗). For the proof, the weighted L2-energy method will be employed with suitable weight, similar to that in Todorova and Yordanov [Y. Todorova, B. Yordanov, Critical exponent for a nonlinear wave equation with damping, J. Differential Equations 174 (2001) 464–489]

Asymptotic behaviors of solutions for time dependent damped wave equations

AbstractIn this paper, we consider the Cauchy problem for the wave equation with time dependent damping b(t)ut and absorbed semilinear term |u|ρ−1u. Here, b(t)=b0(1+t)−β with −1<β<1 and b0>0. Using the weighted energy method, we obtain the L1 and L2 decay rates of the solution, which coincide to those for self-similar solutions to the corresponding parabolic equation when 1<ρ<ρF(N):=1+2N

Boundary Effect on Asymptotic Behaviour of Solutions to the p-System with Linear Damping

AbstractWe consider the asymptotic behaviour of solutions to the p-system with linear damping on the half-line R+=(0, ∞),vt−ux=0,ut+p(v)x=−αu, with the Dirichlet boundary condition u|x=0=0 or the Neumann boundary condition ux|x=0=0. The initial date (v0, u0)(x) has the constant state (v+, u+) at x=∞. L. Hsiao and T.-P. Liu [Commun. Math. Phys.143 (1992), 599–605] have shown that the solution to the corresponding Cauchy problem behaves like diffusion wave, and K. Nishihara [J. Differential Equations131 (1996), 171–188; 137 (1997), 384–395] has proved its optimal convergence rate. Our main concern in this paper is the boundary effect. In the case of null-Dirichlet boundary condition on u, the solution (v, u) is proved to tend to (v+, 0) as t tends to infinity. Its optimal convergence rate is also obtained by using the Green function of the diffusion equation with constant coefficients. In the case of null-Neumann boundary condition on u, v(0, t) is conservative and v(0, t)≡v0(0) by virtue of the first equation, so that v(x, t) is expected to tend to the diffusion wave v(x, t) connecting v0(0) and v+. In fact the solution (v, u)(x, t) is proved to tend to (v(x, t), 0). In the special case v0(0)=v+, the optimal convergence rate is also obtained. However, this is not known in the case v0(0)≠v+

Decay properties of solutions to the Cauchy problem for the damped wave equation with absorption

AbstractWe consider the Cauchy problem for the damped wave equation with absorptionutt−Δu+ut+|u|p−1u=0,(t,x)∈R+×RN. The behavior of u as t→∞ is expected to be the same as that for the corresponding heat equationϕt−Δϕ+|ϕ|p−1ϕ=0,(t,x)∈R+×RN, which has the similarity solution wa(t,x) with the form t−1/(p−1)f(x/t) depending on a=lim|x|→∞|x|2/(p−1)f(x)⩾0 provided that p is less than the Fujita exponent pc(N):=1+2/N. In this paper, as a first step, if 1<p<pc(N) and the data (u0,u1)(x) decays exponentially as |x|→∞ without smallness condition, the solution is shown to decay with rates as t→∞,(∗)(‖u(t)‖L2,‖u(t)‖Lp+1,‖∇u(t)‖L2)=O(t−1p−1+N4,t−1p−1+N2(p+1),t−1p−1−12+N4), those of which seem to be reasonable, because the similarity solution wa(t,x) has the same decay rates as (∗). For the proof, the weighted L2-energy method will be employed with suitable weight, similar to that in Todorova and Yordanov [Y. Todorova, B. Yordanov, Critical exponent for a nonlinear wave equation with damping, J. Differential Equations 174 (2001) 464–489]

Transcutaneous Electrical Nerve Stimulation on the PC-5 and PC-6 Points Alleviated Hypotension after Epidural Anaesthesia, Depending on the Stimulus Frequency

Neuraxial blockade causes arterial hypotension. Transcutaneous electrical nerve stimulation (TENS) at the Neiguan (PC-6) and Jianshi (PC-5) reduces the severity of hypotension after spinal anaesthesia, but did not clarify the optimal stimulus frequency. We hypothesized that the stimulus frequency of TENS at the PC-6 and PC-5 points would influence the severity of hypotension after epidural anaesthesia. 65 ASA I or II male patients presenting for inguinal hernia repair were randomized to five groups: the control group received no treatment; the 2 Hz, 10 Hz, 20 Hz, and 40 Hz groups received TENS at a frequency of 2 Hz, 10 Hz, 20 Hz, and 40 Hz, respectively. The lowest SBP was significantly higher in the 40 Hz group [the control, 84 (74–110) mmHg; the 2 Hz, 96 (62–116) mmHg; the 10 Hz, 100 (68–110) mmHg; the 20 Hz, 96 (64–115) mmHg; the 40 Hz, 104 (75–140) mmHg: P = 0.004]. Significantly less patients experienced hypotension in the 40 Hz group [the control, 78%; the 2 Hz, 43%; the 10 Hz, 38%; the 20 Hz, 38%; the 40 Hz, 8%: P = 0.008]. TENS on the PC-6 and PC-5 points reduced the severity and incidence of hypotension after epidural anaesthesia, depending on the stimulus frequency

In vivo imaging models of bone and brain metastases and pleural carcinomatosis with a novel human EML4-ALK lung cancer cell line

がん進展制御研究所EML4-ALK lung cancer accounts for approximately 3-7% of non-small-cell lung cancer cases. To investigate the molecular mechanism underlying tumor progression and targeted drug sensitivity/resistance in EML4-ALK lung cancer, clinically relevant animal models are indispensable. In this study, we found that the lung adenocarcinoma cell line A925L expresses an EML4-ALK gene fusion (variant 5a, E2:A20) and is sensitive to the ALK inhibitors crizotinib and alectinib. We further established highly tumorigenic A925LPE3 cells, which also have the EML4-ALK gene fusion (variant 5a) and are sensitive to ALK inhibitors. By using A925LPE3 cells with luciferase gene transfection, we established in vivo imaging models for pleural carcinomatosis, bone metastasis, and brain metastasis, all of which are significant clinical concerns of advanced EML4-ALK lung cancer. Interestingly, crizotinib caused tumors to shrink in the pleural carcinomatosis model, but not in bone and brain metastasis models, whereas alectinib showed remarkable efficacy in all three models, indicative of the clinical efficacy of these ALK inhibitors. Our in vivo imaging models of multiple organ sites may provide useful resources to analyze further the pathogenesis of EML4-ALK lung cancer and its response and resistance to ALK inhibitors in various organ microenvironments. © 2015 The Authors

Heterometallic Benzenehexathiolato Coordination Nanosheets: Periodic Structure Improves Crystallinity and Electrical Conductivity.

Funder: White Rock FoundationFunder: Jardine FoundationFunder: Cambridge Commonwealth European and International TrustCoordination nanosheets are an emerging class of 2D, bottom-up materials having fully π-conjugated, planar, graphite-like structures with high electrical conductivities. Since their discovery, great effort has been devoted to expand the variety of coordination nanosheets; however, in most cases, their low crystallinity in thick films hampers practical device applications. In this study, mixtures of nickel and copper ions are employed to fabricate benzenehexathiolato (BHT)-based coordination nanosheet films, and serendipitously, it is found that this heterometallicity preferentially forms a structural phase with improved film crystallinity. Spectroscopic and scattering measurements provide evidence for a bilayer structure with in-plane periodic arrangement of copper and nickel ions with the NiCu2 BHT formula. Compared with homometallic films, heterometallic films exhibit more crystalline microstructures with larger and more oriented grains, achieving higher electrical conductivities reaching metallic behaviors. Low dependency of Seebeck coefficient on the mixing ratio of nickel and copper ions supports that the large variation in the conductivity data is not caused by change in the intrinsic properties of the films. The findings open new pathways to improve crystallinity and to tune functional properties of 2D coordination nanosheets.EPSRC-JSPS core-to-core program (EP/S030662/1, JPJSCCA20190005