The Experts below are selected from a list of 24768 Experts worldwide ranked by ideXlab platform
Motohiro Sobajima - One of the best experts on this subject based on the ideXlab platform.
-
life span of solutions to semiLinear wave equation with time dependent critical damping for specially localized initial data
Mathematische Annalen, 2018Co-Authors: Masahiro Ikeda, Motohiro SobajimaAbstract:This paper is concerned with the blowup phenomena for initial value problem of semiLinear wave equation with critical time-dependent damping term DW $$\begin{aligned} {\left\{ \begin{array}{ll} \partial _t^2 u(x,t) -\Delta u(x,t) + \dfrac{\mu }{1+t}\partial _t u(x,t)=|u(x,t)|^p, &{} (x,t)\in \mathbb {R}^N \times (0,T),\\ u(x,0)=\varepsilon f(x),&{} x\in \mathbb {R}^N,\\ \partial _t u(x,0)=\varepsilon g(x),&{} x\in \mathbb {R}^N, \end{array}\right. } \end{aligned}$$ where $$N\in \mathbb {N},$$ $$\mu \in [0,\frac{N^2+N+2}{N+2})$$ and $$\varepsilon >0$$ is a parameter describing the smallness of initial data. Given data f, g are compactly supported in a small area inside of the unit ball. The result is the upper bound of lifespan of solution u with respects to the small parameter $$\varepsilon $$ when $$p_F(N)\le p\le p_0(N+\mu ),$$ where $$p_F(N)$$ denotes the Fujita exponent for the nonLinear heat equations and $$p_0(n)$$ denotes the Strauss exponent for nonLinear wave equation (DW) in n-dimension with $$\mu =0.$$ Consequently, by connecting the result of D’Abbicco–Lucente–Reissig (J Differ Equ 259:5040–5073, 2015), our result clarifies the threshold exponent $$p_0(N+\mu )$$ for dividing blowup phenomena and global existence of small solutions when $$N=3.$$ The crucial idea is to construct suitable test functions satisfying the Conjugate Linear equation $$\partial _t^2\Phi -\Delta \Phi -\partial _t(\frac{\mu }{1+t}\Phi )=0$$ of (DW) including the Gauss hypergeometric functions; note that the construction of test functions is different from Zhou–Han (Commun Partial Differ Equ 39:439–451, 2014).
-
life span of solutions to semiLinear wave equation with time dependent critical damping for specially localized initial data
arXiv: Analysis of PDEs, 2017Co-Authors: Masahiro Ikeda, Motohiro SobajimaAbstract:This paper is concerned with the blowup phenomena for initial value problem of semiLinear wave equation with critical time-dependent damping term (DW). The result is the sharp upper bound of lifespan of solution with respect to the small parameter $\ep$ when $p_F(N)\leq p\leq p_0(N+\mu)$, where $p_F(N)$ denotes the Fujita exponent for the nonLinear heat equations and $p_0(n)$ denotes the Strauss exponent for nonLinear wave equation in $n$-dimension with $\mu=0$. Consequently, by connecting the result of D'Abbicco--Lucente--Reissig 2015, our result clarifies the threshold exponent $p_0(N+\mu)$ for dividing blowup phenomena and global existence of small solutions when $N=3$. The crucial idea is to construct suitable test functions satisfying the Conjugate Linear equation $\pa_t^2\Phi-\Delta \Phi-\pa_t(\frac{\mu}{1+t}\Phi)=0$ of (DW) including the Gauss hypergeometric functions; note that the construction of test functions is different from Zhou--Han in 2014.
Masahiro Ikeda - One of the best experts on this subject based on the ideXlab platform.
-
life span of solutions to semiLinear wave equation with time dependent critical damping for specially localized initial data
Mathematische Annalen, 2018Co-Authors: Masahiro Ikeda, Motohiro SobajimaAbstract:This paper is concerned with the blowup phenomena for initial value problem of semiLinear wave equation with critical time-dependent damping term DW $$\begin{aligned} {\left\{ \begin{array}{ll} \partial _t^2 u(x,t) -\Delta u(x,t) + \dfrac{\mu }{1+t}\partial _t u(x,t)=|u(x,t)|^p, &{} (x,t)\in \mathbb {R}^N \times (0,T),\\ u(x,0)=\varepsilon f(x),&{} x\in \mathbb {R}^N,\\ \partial _t u(x,0)=\varepsilon g(x),&{} x\in \mathbb {R}^N, \end{array}\right. } \end{aligned}$$ where $$N\in \mathbb {N},$$ $$\mu \in [0,\frac{N^2+N+2}{N+2})$$ and $$\varepsilon >0$$ is a parameter describing the smallness of initial data. Given data f, g are compactly supported in a small area inside of the unit ball. The result is the upper bound of lifespan of solution u with respects to the small parameter $$\varepsilon $$ when $$p_F(N)\le p\le p_0(N+\mu ),$$ where $$p_F(N)$$ denotes the Fujita exponent for the nonLinear heat equations and $$p_0(n)$$ denotes the Strauss exponent for nonLinear wave equation (DW) in n-dimension with $$\mu =0.$$ Consequently, by connecting the result of D’Abbicco–Lucente–Reissig (J Differ Equ 259:5040–5073, 2015), our result clarifies the threshold exponent $$p_0(N+\mu )$$ for dividing blowup phenomena and global existence of small solutions when $$N=3.$$ The crucial idea is to construct suitable test functions satisfying the Conjugate Linear equation $$\partial _t^2\Phi -\Delta \Phi -\partial _t(\frac{\mu }{1+t}\Phi )=0$$ of (DW) including the Gauss hypergeometric functions; note that the construction of test functions is different from Zhou–Han (Commun Partial Differ Equ 39:439–451, 2014).
-
life span of solutions to semiLinear wave equation with time dependent critical damping for specially localized initial data
arXiv: Analysis of PDEs, 2017Co-Authors: Masahiro Ikeda, Motohiro SobajimaAbstract:This paper is concerned with the blowup phenomena for initial value problem of semiLinear wave equation with critical time-dependent damping term (DW). The result is the sharp upper bound of lifespan of solution with respect to the small parameter $\ep$ when $p_F(N)\leq p\leq p_0(N+\mu)$, where $p_F(N)$ denotes the Fujita exponent for the nonLinear heat equations and $p_0(n)$ denotes the Strauss exponent for nonLinear wave equation in $n$-dimension with $\mu=0$. Consequently, by connecting the result of D'Abbicco--Lucente--Reissig 2015, our result clarifies the threshold exponent $p_0(N+\mu)$ for dividing blowup phenomena and global existence of small solutions when $N=3$. The crucial idea is to construct suitable test functions satisfying the Conjugate Linear equation $\pa_t^2\Phi-\Delta \Phi-\pa_t(\frac{\mu}{1+t}\Phi)=0$ of (DW) including the Gauss hypergeometric functions; note that the construction of test functions is different from Zhou--Han in 2014.
Sobajima Motohiro - One of the best experts on this subject based on the ideXlab platform.
-
Life-span of solutions to semiLinear wave equation with time-dependent critical damping for specially localized initial data
2017Co-Authors: Ikeda Masahiro, Sobajima MotohiroAbstract:This paper is concerned with the blowup phenomena for initial value problem of semiLinear wave equation with critical time-dependent damping term (DW). The result is the sharp upper bound of lifespan of solution with respect to the small parameter $\ep$ when $p_F(N)\leq p\leq p_0(N+\mu)$, where $p_F(N)$ denotes the Fujita exponent for the nonLinear heat equations and $p_0(n)$ denotes the Strauss exponent for nonLinear wave equation in $n$-dimension with $\mu=0$. Consequently, by connecting the result of D'Abbicco--Lucente--Reissig 2015, our result clarifies the threshold exponent $p_0(N+\mu)$ for dividing blowup phenomena and global existence of small solutions when $N=3$. The crucial idea is to construct suitable test functions satisfying the Conjugate Linear equation $\pa_t^2\Phi-\Delta \Phi-\pa_t(\frac{\mu}{1+t}\Phi)=0$ of (DW) including the Gauss hypergeometric functions; note that the construction of test functions is different from Zhou--Han in 2014.Comment: arXiv admin note: text overlap with arXiv:1709.0440
Joon Ho Cho - One of the best experts on this subject based on the ideXlab platform.
-
capacity of second order cyclostationary complex gaussian noise channels
IEEE Transactions on Communications, 2012Co-Authors: Byung Wook Han, Joon Ho ChoAbstract:In this paper, we derive the capacity of a continuous-time, single-input single-output (SISO), frequency-selective, band-limited, Linear time-invariant (LTI) channel, whose output is corrupted by a second-order cyclostationary (SOCS) complex Gaussian noise. By using a pair of invertible, Linear-Conjugate Linear time-varying operators called a properizing FREquency SHift (p-FRESH) vectorizer and a p-FRESH scalarizer, it is shown that, whether the complex noise is proper or improper, the SISO channel can always be converted to an equivalent multiple-input multiple-output (MIMO) LTI channel whose output is now corrupted by a proper-complex vector wide-sense stationary noise. A variational problem is then formulated in the frequency domain to find the optimal input distribution that maximizes the throughput of the equivalent MIMO channel. It turns out that the optimal input to the SISO channel, obtained through a procedure similar to the water filling, is an SOCS complex Gaussian random process with the same cycle period as the noise. It is shown that this procedure, named cyclic water filling, significantly outperforms ordinary water filling by effectively utilizing the spectral correlation of the cyclostationary noise.
Lajos Molnar - One of the best experts on this subject based on the ideXlab platform.
-
MAPS ON POSITIVE OPERATORS PRESERVING
2016Co-Authors: Lajos MolnarAbstract:Abstract. Let H be a complex Hilbert space. Denote by B(H)+ the set of all positive bounded Linear operators on H. A bijective map φ: B(H)+ → B(H)+ is said to preserve Lebesgue decom-positions in both directions if for any quadruple A,B,C,D of positive operators, B = C +D is an A-Lebesgue decomposition of B if and only if φ(B) = φ(C)+φ(D) is a φ(A)-Lebesgue decomposition of φ(B). It is proved that every such transformation φ is of the form φ(A) = SAS ∗ (A ∈ B(H)+) for some invertible bounded Linear or Conjugate-Linear operator S on H