We concern the blow up problem to the scale invariant damping wave equations with sub-Strauss exponent. This problem has been studied by Lai, Takamura and Wakasa ([5]) and Ikeda and Sobajima [4] recently. In present paper, we extend the blowup exponent from p F (n) ≤ p < p S (n + 2µ) to 1 < p < p S (n + µ) without small restriction on µ. Moreover, the upper bound of lifespan is derived with uniformly estimate T (ε) ≤ Cε −2p(p−1)/γ (p,n+2µ) . This result extends the blowup result of semilinear wave equation and shows the wave-like behavior of scale invariant damping wave equation's solution even with large µ > 1.

No abstract

We consider the Cauchy problem for the damped wave equation with space-time dependent potential b(t, x) and absorbing semilinear termUsing the weighted energy method, we can obtain the L 2 decay rate of the solution, which is almost optimal in the case ρ > ρ c (N, α, β) := 1 + 2/(N − α). Combining this decay rate with the result that we got in the paper [J. Lin, K. Nishihara, J. Zhai, L 2 -estimates of solutions for damped wave equations with space-time dependent damping term, J. Differential Equations 248 (2010) 403-422], we believe that ρ c (N, α, β) is a critical exponent. Note that when α = β = 0, ρ c (N, α, β) coincides to the Fujita exponent ρ F (N) := 1 + 2/N. The new points include the estimate in the supercritical exponent and for not necessarily compactly supported data.

Purpose. Our aim is to conduct analysis and comparison of some methods commonly used to measure the volume of hematoma, for example, slice method, voxelization method, and 3D-Slicer software method (projection method). Method. In order to validate the accuracy of the slice method, voxelization method, and 3D-Slicer method, these three methods were first applied to measure two known volumetric models, respectively. Then, a total of 198 patients diagnosed with spontaneous intracerebral hemorrhage (ICH) were recruited. The patients were split into 3 different groups based on the hematoma size: group 1: volume<10 ml (n=89), group 2: volume between 10 and 20 ml (n=59), and group 3: volume>20 ml (n=50). And the shape of the hematoma was classed into regular (round to ellipsoid) with smooth margins (n=76), irregular with frayed margins (n=85), and multilobular (n=37). The slice method, voxelization method, and 3D-Slicer method were adopted to measure the volume of hematoma, respectively, considering the nonclosed models and the models which may contain inaccurate normal information during CT scan. Moreover, the results were compared with the 3D-Slicer method for closed models. Results. There was a significant estimation error (P<0.05) using these three methods to calculate the volume of the closed hematoma model. The estimated hematoma volume was calculated to be 14.2086743±0.900559087 ml, 14.2119130±0.900851812 ml, and 14.2123825±0.900835916 ml using slice method 1, slice method 2, and the voxelization method, respectively, compared to 14.212656±0.900992371 ml using the 3D-Slicer method. The mean estimation error was -0.00398172 ml, -0.00074303 ml, and -0.00027354 ml caused by slice method 1, slice method 2, and voxelization method, respectively. There was a significant estimation error (P<0.05), applying these three methods to calculate the volume of the nonclosed hematoma model. The estimated hematoma volume was calculated to be 14.1928246±0.902210314 ml using the 3D-Slicer method. The mean estimation error was calculated to be -0.00402121 ml, -0.00078237 ml, -0.00031288 ml, and -0.01983136 ml using slice method 1, slice method 2, voxelization method, and 3D-Slicer method, respectively. Conclusions. The 3D-Slicer software method is considered as a stable and capable method of high precision for the calculation of a closed hematoma model with correct normal direction, while it would be inappropriate for the nonclosed model nor the model with incorrect normal direction. The slice method and voxelization method can be the supplement and improvement of the 3D-Slicer software method, for the purpose of achieving precision medicine.

The blow up problem of the semilinear scale-invariant damping wave equation with critical Strauss type exponent is investigated. The life span is shown to be:n+2 . This result completes our previous study [12] on the sub-Strauss type exponent p < p S (n + µ). Our novelty is to construct the suitable test function from the modified Bessel function. This approach might be also applied to the other type damping wave equations.