我刚刚尝试了PendingIntent.FLAG_CANCEL_CURRENT，似乎对我有效

public void showNotification(String header,String message){

            // define sound URI, the sound to be played when there's a notification
            Uri soundUri = RingtoneManager.getDefaultUri(RingtoneManager.TYPE_NOTIFICATION);

            Intent intent = new Intent(this, MainActivity.class);

            //PendingIntent.FLAG_CANCEL_CURRENT will bring the app back up again
            PendingIntent pIntent = PendingIntent.getActivity(MainActivity.this,PendingIntent.FLAG_CANCEL_CURRENT, intent, 0);

            Notification mNotification = new Notification.Builder(this)
                .setContentTitle(header)
                .setContentText(message)
                .setSmallIcon(R.drawable.ic_launcher)
                .setContentIntent(pIntent)
                .setSound(soundUri)

                .addAction(R.drawable.ic_launcher, "View", pIntent)
                .addAction(0, "Remind", pIntent)
                .setOngoing(true)//optional
                .build();

            NotificationManager notificationManager = (NotificationManager) getSystemService(NOTIFICATION_SERVICE);

            notificationManager.notify(0, mNotification);
}

在MainActivity的清单声明中使用android:launchMode="singleTop"

在做了大量的搜索之后，对我来说唯一有效的解决方案是：

NotificationCompat.Builder builder = new NotificationCompat.Builder(this).set...(...).set...(..);

Intent resultIntent = new Intent(this, MainClass.class);
resultIntent.setAction("android.intent.action.MAIN");
resultIntent.addCategory("android.intent.category.LAUNCHER");

PendingIntent resultPendingIntent = PendingIntent.getActivity(this, 0, resultIntent, PendingIntent.FLAG_UPDATE_CURRENT);

builder.setContentIntent(resultPendingIntent);
NotificationManager mNotificationManager = (NotificationManager) getSystemService(Context.NOTIFICATION_SERVICE);
mNotificationManager.notify(0, builder.build());

这将打开您当前的Activity，而不创建另一个Activity！

@帕特里克提供的解决方案取自问题并作为答案添加：

NotificationCompat.Builder mBuilder = 
                new NotificationCompat.Builder(this) 
                .setSmallIcon(R.drawable.logo)    
                .setContentTitle(getString(R.string.txt)) 
                .setContentText(getString(R.string.txt)); 

        mBuilder.setOngoing(true); 

        // Creates an explicit intent for an Activity in your app 
        Intent resultIntent = new Intent(this, Activity.class); 

        resultIntent.setFlags(Intent.FLAG_ACTIVITY_CLEAR_TOP | Intent.FLAG_ACTIVITY_SINGLE_TOP); 

        // The stack builder object will contain an artificial back stack for the 
        // started Activity. 
        // This ensures that navigating backward from the Activity leads out of 
        // your application to the Home screen. 
        TaskStackBuilder stackBuilder = TaskStackBuilder.from(this); 
        // Adds the back stack for the Intent (but not the Intent itself) 
        stackBuilder.addParentStack(Activity.class); 
        // Adds the Intent that starts the Activity to the top of the stack 
        stackBuilder.addNextIntent(resultIntent); 
        PendingIntent resultPendingIntent = PendingIntent.getActivity(this, 0, resultIntent, 0); 
        mBuilder.setContentIntent(resultPendingIntent); 
        NotificationManager mNotificationManager = 
            (NotificationManager) getSystemService(Context.NOTIFICATION_SERVICE); 
        // mId allows you to update the notification later on. 
        mNotificationManager.notify(mId, mBuilder.getNotification());