Consider the following two dimensional system $\dot{x_1}=x_2-x_1^3$ , $\dot{x_2}=x_1+x_2-2x_2^3$
Using the Lyapunov function $\operatorname{V}(x)=\frac 12x_1^2+\frac 12x_2^2$ and the Young inequalities $2x_1x_2 \le x_1^2+x_2^2$ ,$2x_1^2x_2^2 \le x_1^4+x_2^4$
prove that every solution of the dynamical system is bounded.
I know that we have to show that $\operatorname{V'}(x)\le-c_1\|x(t)\|^r+L$ for some positive constants r,L. Another way to go is by showing that $\operatorname{V'}(x)\le-c_1\operatorname{V}^q$ for some positive constants $c_1\ge0,q \ge1$
However I cant find a way to transform the inequality in any of these two ways , so if anyone could give me a hand I would really appreciate it. Thank you in advance!