Pointwise estimates for solutions of fractal Burgers equation (Q324110)

Suppose \(d \in \mathbb{N}, \alpha \in (1,2)\) and \(q_0=(\alpha-1)/d\). The aim of this article is to describe estimates and asymptotics of solutions to the fractal Burgers equation NEWLINE\[NEWLINEu_t-\Delta^{\alpha/2}u+b\cdot \nabla(u|u|^q)=0, \; u(0,x)=u_0(x). \eqno{(1)} NEWLINE\]NEWLINE Here \(q \geq q_0\) and \(b\in \mathbb{R}^d\) is a constant vector, \(u_0\in L^1\) and \(u_0\geq 0\). Then, according to \textit{P. Biler} et al. [Stud. Math. 148, No. 2, 171--192 (2001; Zbl 0990.35023)] the solution \(u(t,x)\) is also non-negative. The pseudo-differential operator \(\Delta^{\alpha/2}\) is defined with the aid of the Fourier transform.NEWLINENEWLINEThe basic results of the article are contained in the following statements:NEWLINENEWLINETheorem 1. Let one of the conditionsNEWLINENEWLINE1). \(u_0 \in L^1(\mathbb{R}^d)\;, u_0\geq 0\) for \(q=q_0\);NEWLINENEWLINE2). \(u_0 \in L^1(\mathbb{R}^d)\bigcap L^{\infty}(\mathbb{R}^d)\;, u_0\geq 0\) for \(q>q_0\)NEWLINENEWLINEhold. Then, the solution \(u(t, x)\) of (1) satisfies NEWLINE\[NEWLINE\frac{1}{C}(P_tu_0)(x)\leq u(t,x) \leq C(P_tu_0)(x) NEWLINE\]NEWLINE for some \(C=C(d,\alpha,u_0)>1.\)NEWLINENEWLINETheorem 1 gives the following corollaries:NEWLINENEWLINE1). The solution of (1) is strictly positive whenever \(u_0\geq 0\) and \(\|u_0\|_1> 0\).NEWLINENEWLINE2). For every \(u_0 \in L^1(\mathbb{R}^d)\), \(\lim_{t\to\infty}t^{d(1-1/p)/{\alpha}}\|(P_tu_0)(\cdot)-Mp(t,\cdot)\|_p=0\) for each \( p \in [1,\infty)\).NEWLINENEWLINEIn conclusion of Section 5. the authors give a description of the asymptotic behavior of the solution \(u(t, x)\).