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Động lực học của phương trình phi tuyến hyperbolic loại Kirchhoff
Tóm tắt
Trong bài báo này, chúng tôi nghiên cứu bài toán giá trị biên ban đầu của phương trình Kirchhoff hyperbolic quan trọng
$$\begin{aligned} u_{tt}-\left( a \int _\text{\O}mega |\nabla u|^2 \mathrm {d}x +b\right) \Delta u = \lambda u+ |u|^{p-1}u , \end{aligned}$$
trong đó a, $$b>0$$, $$p>1$$, $$\lambda \in {\mathbb {R}}$$ và năng lượng ban đầu là rất lớn. Chúng tôi chứng minh một số định lý mới về động lực học, chẳng hạn như tính giới hạn hoặc gia tăng hữu hạn thời gian của nghiệm dưới các giá trị khác nhau của a, b, $$\lambda$$ và dữ liệu ban đầu trong các trường hợp sau: (i) $$11/\Lambda$$, (iii) $$p=3$$, $$a \le 1/\Lambda$$ và $$\lambda b\lambda _1$$, (v) $$p>3$$ và $$\lambda \le b\lambda _1$$, (vi) $$p>3$$ và $$\lambda > b\lambda _1$$, trong đó $$\lambda _1 = \inf \left\{ \Vert \nabla u\Vert ^2_2 :~ u\in H^1_0(\text{\O}mega )\ \mathrm{và}\ \Vert u\Vert _2 =1\right\} $$ và $$\Lambda = \inf \left\{ \Vert \nabla u\Vert ^4_2 :~ u\in H^1_0(\text{\O}mega )\ \mathrm{và}\ \Vert u\Vert _4 =1\right\} $$.
Hơn nữa, chúng tôi chứng minh tính bất biến của một số tập hợp ổn định và không ổn định của nghiệm đối với a, b và $$\lambda$$ phù hợp, và đưa ra các điều kiện đủ của dữ liệu ban đầu để tạo ra một vùng chân không của nghiệm. Do hiệu ứng không địa phương do điều khoản tích phân vi phân không địa phương gây ra, chúng tôi chỉ ra nhiều sự khác biệt thú vị giữa hiện tượng gia tăng của bài toán trong trường hợp $$a>0$$ và $$a=0$$.
Từ khóa
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