Add authentication check and logout functionality in app.vue

server/api/auth/check.get.js

@@ -0,0 +1,21 @@
+export default defineEventHandler(async (event) => {
+  const token = getCookie(event, 'auth-token')
+  
+  if (!token) {
+    return { authenticated: false }
+  }
+  
+  const session = await useStorage('memory').getItem(`session:${token}`)
+  
+  if (!session) {
+    return { authenticated: false }
+  }
+  
+  // Check if session has expired
+  if (session.expiresAt && new Date() > new Date(session.expiresAt)) {
+    await useStorage('memory').removeItem(`session:${token}`)
+    return { authenticated: false }
+  }
+  
+  return { authenticated: true }
+})