gradient descent relaxation dynamics of a Euler-Lagrange equation

Question

To minimize the functional $\int{L(u)}dx$, where L=$u_x^2-u^2$. Though its Euler-Lagrange equation is easy to solve, I want to turn it into solving a gradient descent problem as a general method, for the Euler-Lagrange will be hard to solve if the functional is replaced with a complicated one. I propose to solve the gradient descent dynamics $u’(t)=-\delta{L}/\delta{u}$, just a relaxation dynamics of the original Euler-Lagrange statics. I hope, as t goes to inf, the solution to this gradient descent dynamics coincides with that to the original Euler-Lagrange eq. However, the solution is diverged for the obvious reason that $-u^2$ is concave down in u. How to resolve/rewrite this relaxation dynamics for my purpose?