Just writing to confirm Håvard H. Garnes' workaround:

>This is because of new faulty qt4.4 from hardy backports. Use synaptic, remove all qt, libqt, libqtcore and everything else qt (lyx, skype and 
>stellarium will follow). Then remove hardy-backports from the sources and resync. Then install libqt4-core and libqt4-gui at version 4.3. Then 
>lyx and everythong else. Now everything should be fine.

I have now opened and closed several documents, and LyX appears stable. The ~/.config/Trolltech.conf that lyx creates under QT 4.3 looks as follows (I have no idea whether this is useful, but here goes):

--- .config/Trolltech.conf.lyx-1.5.3-1.qt4.3    2008-05-09 19:31:26.000000000 +0200
+++ .config/Trolltech.conf.lyx-1.5.3-1.qt4.4    2008-05-09 19:12:47.000000000 +0200
@@ -1,18 +1,20 @@
-[Qt%20Factory%20Cache%204.3]
-com.trolltech.Qt.QIconEngineFactoryInterface%3A\usr\lib\qt4\plugins\iconengines\libqsvg.so=2008-04-09T19:41:59
-com.trolltech.Qt.QIconEngineFactoryInterfaceV2%3A\usr\lib\qt4\plugins\iconengines\libqsvg.so=2008-04-09T19:41:59, svg
-com.trolltech.Qt.QImageIOHandlerFactoryInterface%3A\usr\lib\qt4\plugins\imageformats\libqgif.so=2008-04-09T19:41:59, gif
-com.trolltech.Qt.QImageIOHandlerFactoryInterface%3A\usr\lib\qt4\plugins\imageformats\libqjpeg.so=2008-04-09T19:41:59, jpeg, jpg
-com.trolltech.Qt.QImageIOHandlerFactoryInterface%3A\usr\lib\qt4\plugins\imageformats\libqmng.so=2008-04-09T19:41:59, mng
-com.trolltech.Qt.QImageIOHandlerFactoryInterface%3A\usr\lib\qt4\plugins\imageformats\libqsvg.so=2008-04-09T19:41:59, svg
-com.trolltech.Qt.QImageIOHandlerFactoryInterface%3A\usr\lib\qt4\plugins\imageformats\libqtiff.so=2008-04-09T19:41:59, tiff, tif
-com.trolltech.Qt.QInputContextFactoryInterface%3A\usr\lib\qt4\plugins\inputmethods\libqimsw-multi.so=2008-04-09T19:41:59, imsw-multi
+[Qt%20Plugin%20Cache%204.4.false]
+usr\lib\qt4\plugins\inputmethods\libqimsw-multi.so=40400, 0, x86_64 Linux g++-4 full-config, 2008-05-06T23:17:16
+usr\lib\qt4\plugins\imageformats\libqgif.so=40400, 0, x86_64 Linux g++-4 full-config, 2008-05-06T23:17:16
+usr\lib\qt4\plugins\imageformats\libqico.so=40400, 0, x86_64 Linux g++-4 full-config, 2008-05-06T23:17:16
+usr\lib\qt4\plugins\imageformats\libqjpeg.so=40400, 0, x86_64 Linux g++-4 full-config, 2008-05-06T23:17:16
+usr\lib\qt4\plugins\imageformats\libqmng.so=40400, 0, x86_64 Linux g++-4 full-config, 2008-05-06T23:17:16
+usr\lib\qt4\plugins\imageformats\libqsvg.so=40400, 0, x86_64 Linux g++-4 full-config, 2008-05-06T23:17:31
+usr\lib\qt4\plugins\imageformats\libqtiff.so=40400, 0, x86_64 Linux g++-4 full-config, 2008-05-06T23:17:16
+usr\lib\qt4\plugins\iconengines\libqsvgicon.so=40400, 0, x86_64 Linux g++-4 full-config, 2008-05-06T23:17:31
 
-[Qt%20Plugin%20Cache%204.3.debug]
-usr\lib\qt4\plugins\iconengines\libqsvg.so=40304, 1, x86_64 Linux g++-4 full-config, 2008-04-09T19:41:59
-usr\lib\qt4\plugins\imageformats\libqgif.so=40304, 1, x86_64 Linux g++-4 full-config, 2008-04-09T19:41:59
-usr\lib\qt4\plugins\imageformats\libqjpeg.so=40304, 1, x86_64 Linux g++-4 full-config, 2008-04-09T19:41:59
-usr\lib\qt4\plugins\imageformats\libqmng.so=40304, 1, x86_64 Linux g++-4 full-config, 2008-04-09T19:41:59
-usr\lib\qt4\plugins\imageformats\libqsvg.so=40304, 1, x86_64 Linux g++-4 full-config, 2008-04-09T19:41:59
-usr\lib\qt4\plugins\imageformats\libqtiff.so=40304, 1, x86_64 Linux g++-4 full-config, 2008-04-09T19:41:59
-usr\lib\qt4\plugins\inputmethods\libqimsw-multi.so=40304, 1, x86_64 Linux g++-4 full-config, 2008-04-09T19:41:59
+[Qt%20Factory%20Cache%204.4]
+com.trolltech.Qt.QInputContextFactoryInterface%3A\usr\lib\qt4\plugins\inputmethods\libqimsw-multi.so=2008-05-06T23:17:16, imsw-multi
+com.trolltech.Qt.QImageIOHandlerFactoryInterface%3A\usr\lib\qt4\plugins\imageformats\libqgif.so=2008-05-06T23:17:16, gif
+com.trolltech.Qt.QImageIOHandlerFactoryInterface%3A\usr\lib\qt4\plugins\imageformats\libqico.so=2008-05-06T23:17:16, ico
+com.trolltech.Qt.QImageIOHandlerFactoryInterface%3A\usr\lib\qt4\plugins\imageformats\libqjpeg.so=2008-05-06T23:17:16, jpeg, jpg
+com.trolltech.Qt.QImageIOHandlerFactoryInterface%3A\usr\lib\qt4\plugins\imageformats\libqmng.so=2008-05-06T23:17:16, mng
+com.trolltech.Qt.QImageIOHandlerFactoryInterface%3A\usr\lib\qt4\plugins\imageformats\libqsvg.so=2008-05-06T23:17:31, svg
+com.trolltech.Qt.QImageIOHandlerFactoryInterface%3A\usr\lib\qt4\plugins\imageformats\libqtiff.so=2008-05-06T23:17:16, tiff, tif
+com.trolltech.Qt.QIconEngineFactoryInterfaceV2%3A\usr\lib\qt4\plugins\iconengines\libqsvgicon.so=2008-05-06T23:17:31, svg
+com.trolltech.Qt.QIconEngineFactoryInterface%3A\usr\lib\qt4\plugins\iconengines\libqsvgicon.so=2008-05-06T23:17:31