PDEs are massively used in the theory of harmonic maps.

My personal favourite is a nice theorem by Lemaire and Sacks-Uhlenbeck.

Theorem. Suppose $M$ is a compact Riemann surface, possibly with boundary, $N \subset \mathbb R^n$ is compact. If $\pi_2(N) = 0$, then any map $u_0: M \to N$ is homotopic to a smooth harmonic map.

The key ingredient of the proof relies on existence and uniqueness of global weak "energy" solutions $u:\ M\times[0,\infty])\to N$ to a nonlinear Cauchy problem for the $L^2$-gradient flow $$\begin{cases} u_t-\triangle_M u=A(u)(\nabla u,\nabla u)_M & \mbox{in }M\times[0,\infty),\\\ u(0)=u_0 & \mbox{on }\partial M\times[0,\infty)\end{cases}$$ which converge to a smooth harmonic map $u_{\infty}:\ M\to N$ as $t\to\infty$.

PDEs are massively used in the theory of harmonic maps.

My personal favourite is a nice theorem by Lemaire and Sacks-Uhlenbeck.

Theorem. Suppose $M$ is a compact Riemann surface, possibly with boundary, $N \subset \mathbb R^n$ is compact. If $\pi_2(N) = 0$, then any map $u_0: M \to N$ is homotopic to a smooth harmonic map.

The key ingredient of the proof relies on existence and uniqueness of global weak "energy" solutions $u:\ M\times[0,\infty])\to N$ to a nonlinear Cauchy problem for the $L^2$-gradient flow $$\begin{cases} u_t-\triangle_M u=A(u)(\nabla u,\nabla u)_M & \mbox{in }M\times[0,\infty),\\\ u=u_0 & \mbox{at }t=0\mbox{ and on }\partial M\times[0,\infty)\end{cases}$$ which converge to a smooth harmonic map $u_{\infty}:\ M\to N$ as $t\to\infty$.