Second critical exponent for a nonlinear nonlocal diffusion equation
The author describes the asymptotic behavior when time goes to \(+\infty \) of the solution \(u\) to the nonlinear and nonlocal equation \(u_{t}=J\ast u-u+u^{p}\) posed in \(\mathbb{R}^{N}\times (0,T)\) and starting from the initial condition \(u(x,0)=\lambda \varphi (x)\) in \(\mathbb{R}^{N}\). Here \(J\) is nonnegative, bounded and satisfies \(\int_{\mathbb{R}^{N}}Jdx=1\), \( \widehat{J}(\xi )=1-A\left| \xi \right| ^{\beta }+o(\left| \xi \right| ^{\beta })\) as \(\left| \xi \right| \rightarrow 0\) for \( 0<\xi \leq 2\) and \(\left| \widehat{J}(\xi )\right| \leq \frac{C}{ \left| \xi \right| ^{m}}\) as \(\left| \xi \right| \rightarrow \infty \) for \(m>N\). The initial data \(\varphi \) is a nonnegative and bounded function in \(\mathbb{R}^{N}\) which satisfies \(\varphi (x)\sim \left| x\right| ^{-a}\) as \(\left| x\right| \rightarrow 0\) and \(\lambda >0\). The main result of the paper proves that if \(p>p_{c}=1+\frac{\beta }{N}\) and \(a\in (0,a^{\ast })\) with \(a^{\ast }=\frac{\beta }{p-1}\) then the solution \(u\) blows up in finite time. On the contrary, if \(a>a^{\ast }\) and \( \lambda \) is small enough the solution \(u\) is global in time. To prove the blowup phenomenon, the author introduces the functions \(v(x,t)=\lambda ^{-1}u(\lambda ^{\theta }x,\lambda ^{1-p}t)\) with \(\theta =\frac{1-p}{\beta } \), \(\rho _{\varepsilon }(x)=e^{-\varepsilon \left| x\right| ^{2}}\), \( \psi _{\varepsilon }(x)=A_{0}\varepsilon ^{\frac{N}{2}}\rho _{\varepsilon }^{l}(x)\) for some constants \(A_{0}\), \(l\) and \(\varepsilon \) and \(F(t)=\int_{ \mathbb{R}^{N}}v(x,t)\psi _{\varepsilon }(x)dx\). Studying these functions, he proves the blowup of \(F\) in finite time under the above-indicated hypotheses. For the proof of the global in time existence result, the author introduces the function \(K_{t}=e^{-t}\sum_{k=1}^{\infty }\frac{t^{k}}{k!} J^{\ast (k)}\) for which he proves that \(\left\| K_{t}\right\| _{q}\leq C(1+t)^{-\frac{N}{\beta }(1-\frac{1}{q})}\) for any \(q\in [ 1,+\infty ]\) and he mainly uses a Banach fixed point theorem.
scientific article; zbMATH DE number 6869198