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تسجيل مستخدم جديدGlobal existence for semilinear damped wave equations in relation with the Strauss conjecture
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We study the global existence of solutions to semilinear wave equations with power-type nonlinearity and general lower order terms on $n$ dimensional nontrapping asymptotically Euclidean manifolds, when $n=3, 4$. In addition, we prove almost global existence with sharp lower bound of the lifespan for the four dimensional critical problem.
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We study the global existence of solutions to semilinear damped wave equations in the scattering case with derivative power-type nonlinearity on (1+3) dimensional nontrapping asymptotically Euclidean manifolds. The main idea is to exploit local energ
y estimate, together with local existence to convert the parameter $mu$ to small one.
It is believed or conjectured that the semilinear wave equations with scattering space dependent damping admit the Strauss critical exponent, see Ikehata-Todorova-Yordanov cite{ITY}(the bottom in page 2) and Nishihara-Sobajima-Wakasugi cite{N2}(conje
cture iii in page 4). In this work, we are devoted to showing the conjecture is true at least when the decay rate of the space dependent variable coefficients before the damping is larger than 2. Also, if the nonlinear term depends only on the derivative of the solution, we may prove the upper bound of the lifespan is the same as that of the solution of the corresponding problem without damping. This shows in another way the lqlq hyperbolicity of the equation.
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In this paper, we study the semilinear wave equations with the inverse-square potential. By transferring the original equation to a fractional dimensional wave equation and analyzing the properties of its fundamental solution, we establish a long-tim
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