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I have many products that have no description but all have short descriptions
How can I copy from short description if description is empty?

You will need to change this file template/catalog/product/view/description.phtml like this:

<?php $_description = $this->getProduct()->getDescription(); ?>
<?php if ($_description): ?>
 <h2><?php echo $this->__('Details') ?></h2>
 <div class="std">
 <?php echo $this->helper('catalog/output')->productAttribute($this->getProduct(), $_description, 'description') ?>
 </div>
<?php else: //adding short description?>
 <h2><?php echo $this->__('Short Details') ?></h2>
 <div class="std">
 <?php echo $_helper->productAttribute($this->getProduct(), $this->getProduct()->getShortDescription(), 'short_description') ?>
 </div>
<?php endif; ?>

This should give you short description if description is empty.

UPDATE

If you want to save short description as description if it is empty in backend then I would suggest to run a simple function like this:

$products = Mage::getModel('catalog/product')->getCollection();
foreach($proudcts as $product)
 if(!$product->getDescription() && $product->getShortDescription())
 $product->setDescription($product->getShortDescription())->save();
 Mage::getSingleton('adminhtml/session')->addSuccess($this->__($product->getId(). "'s description is saved."));
 

Now, how to run this code? You can create a simple module and place above code in indexAction(), then call it.

PS I would still stick with first solution that I provided. That works with not having description in the product. This helps you to know which product doesn't have description and can save later if needed for each product.

I would recommend filtering the collection for performance's sake, and I would put this code into a file in your Magento root (maybe setDescription.php) and run it from either from the command line (php setDescription.php) or the browser (http://yourdevurl/setDescription.php).

<?php

require_once 'app/Mage.php';
Mage::app()->setCurrentStore(Mage_Core_Model_App::ADMIN_STORE_ID);


$productCollection = Mage::getModel('catalog/product')->getCollection()
 ->addAttributeToSelect('short_description')
 ->addAttributeToSelect('description')
 ->addAttributeToFilter('description', array('neq' => NULL));

foreach ($productCollection as $product) 
 $product->setDescription($product->getShortDescription());
 $product->save();