Is it permissible to delay prayer when one experiences a wet dream and is unable to perform ghusl due to embarrassment before one's family and the cold weather?

The one who experiences a wet dream must perform ghusl (ritual bath) and offer the prayer on time. It is not permissible to delay this due to embarrassment from family, as this is a weakness that must be overcome. If ghusl is not possible due to extreme cold and there is a fear of harm, then one must perform tayammum (dry ablution) and pray on time. This is based on the hadith of Amr ibn al-Aas, when he performed tayammum and led the Companions in prayer on a cold night, fearing perishing.