Some Recent Works to One Dimensional Klein-Gordon Equations (Single Case)


In this post, we introduce some recent works to the single nonlinear Klein-Gordon equations in one space dimension. It should be noted that cubic nonlinear Klein-Gordon equations in one space dimension are of special interest, because the large-time behavior of … Continue reading

1차원 클라인 골든 방정식에 대한 고찰 (Time Decay 의 관점에서)


이 쯤 해서 내 나름대로 정리도 할 겸 클라인 골든 방정식(Klein-Gordon Equation)에 대한 포스트를 써 두어 본다. 먼저 1절에서는 $n$ 차원 클라인 골든 방정식에 대한 고전 결과들을 소개하고, 2절에서는 업계(?)에서 주로 쓰는 용어들을 설명한다. 3절에서는 단독(単独) 1차원 클라인 골든 방정식에 … Continue reading

[M1 Seminar II] Week 7 : A Global Existence Theorem for Nonlinear Wave Equations

We consider the following initial value problem \[ y_{tt} – \Delta y = f(Dy,\nabla Dy)\;\;\;\;\text{with}\;\;\;\; y(t=0) = y_0,\; y_t (t=0) = y_1 \tag{P} \] with $f \in C^\infty ( \Bbb R^{(n+1)^2}, \Bbb R)$, $\exists \alpha \in \Bbb N$ such that … Continue reading

[M1 Seminar II] Week 6 : Weighted a priori Estimates for Small Data

For the proof of the global existence theorem of nonlinear wave equations, this a priori estimates is essential as well as the high energy estimates. Also the high energy estimates play an important role in the proof of a priori … Continue reading