On the Cauchy problem for the weakly dissipative modified Camassa–Holm–Novikov equation

The main idea is to build blow-up mechanism for a cubic nonlinear shallow-water model with weakly dissipative term, including two notable integrable systems: the Fokas–Olver–Rosenau–Qiao equation (also known as the modified Camassa–Holm equation) and the Novikov equation. Our blow-up analysis begins by considering two cases: the one is \(2k_1+3k_2 \ne 0\) $2k_1+3k_2\ne 0$ and the other is \(2k_1+3k_2 = 0\) $2k_1+3k_2=0$ . To solve the loss of the conserved quantities \(\mathcal {H}_1[u] =\int _{\mathbb {R}}u^2+u_x^2 \textrm{d}x\) ${\mathcal{H}}_1[u]={\int }_{\mathbb{R}}{u}^2+u_x^2\text{d}x$ and \(\mathcal {H}_2[u]= \int _{\mathbb {R}} u^4+2u^2u_x^2-\frac{1}{3}u_x^4 \textrm{d}x\) ${\mathcal{H}}_2[u]={\int }_{\mathbb{R}}{u}^4+2u^2{u}_x^2-\frac{1}{3}{u}_x^4\text{d}x$ owing to the presence of the weakly dissipative term, we think about establishing relational energy inequalities(cf. Lemma 2.3 and Lemma 3.2). It is hard to directly control the solution u without applying the sign preservation of momentum density y, so we could not directly infer the monotonicity of u according to its dynamics along the characteristics. We solve this problem by taking dynamics of \(uu_x^2-C_1u_x\) $u{u}_x^2-C_1{u}_x$ . Meanwhile, combined with the dynamics of \(u_x\) $u_x$ , we obtain the monotonicity of u and \(u_x\) $u_x$ , which leads to a Riccati dynamics of y and Y (cf. Theorem 3.1 and Theorem 3.2).