Long-time asymptotic behaviors of solutions of \(N\)-dimensional dissipative partial differential equations
The optimal rate of decay for the global solution of the initial value problem for a certain higher-order nonlinear PDE with dissipation \[ \omega_t-\triangle \omega_t+\beta \nabla \omega=\alpha \triangle \omega, \quad \omega(x,0)=\omega_0(x),\quad x \in {\mathbb{R}^N} \] is established. The Fourier splitting technique invented by Maria Shonbek is used. Denote \(l(u)=\int_{\mathbb{R}^N}[|u(x,t)|^2+ |\nabla u(x,t)|^2] dx\). Assuming that long time asymptotic behaviors of solutions of dissipative PDEs satisfy the differential equation \[ \frac{d}{dt} l(u)+2\alpha \int_{\mathbb{R}^N}|(-\triangle)^{\sigma/2}u(x,t)|^2 \;dx= 2\int_{\mathbb{R}^N}u(x,t) f(x,t) dx, \] the author obtains the following main results: (i) If \(u_0(x) \in X,\) then \((1+t)^{n/(2\sigma)}l(u) \leq C,\) (ii) if \(u_0 \in Y\), then \((1+t)^{(n+2)/(2\sigma)}l(u) \leq C,\) (iii) if \(u_0 \in Z\), then \((1+t)^{(n+4)/(2\sigma)}l(u) \leq C,\) (iv) if \(u_0 \in Z_{\infty}\), then \((1+t)^{(n+2\rho)/(2\sigma)}l(u) \leq C,\) when \(\rho \geq 2,\) where \(X,Y,Z,Z_{\infty}\) are certain functional spaces for initial data introduced by the author. (v) There holds the decay result \[ (1+t)^n l(u) \leq C \] if \(u_0 \in L'(\mathbb{R}^N)\cap H^2(\mathbb{R}^N)\). The author extends the obtained results to investigate a wider class of PDEs with dissipation. Some applications of the offered method to a certain class of nonlinear problems are also considered.
dissipative partial differential equations
scientific article; zbMATH DE number 1836304
scientific article