AVDIIVKA, Ukraine — Russian-backed separatists kept up a rocket and artillery attack on this frigid city Wednesday, in a surge in violence that could pose an early and difficult foreign policy challenge to the new Trump administration.

A planned evacuation of Avdiivka, organized by the Ukrainian government, found few takers Wednesday. Only 145 residents chose to board buses that would take them away from the fighting; 88 were children.

Sporadic shelling of Avdiivka, on the front line between separatists and regular Ukrainian forces, had intensified early this week, shortly after President Trump and Russian President Vladimir Putin had their first phone conversation. The sudden eruption in the long-running conflict in eastern Ukraine threatens to put Trump, who has said he wants better relations with Moscow, on the spot.

Analysts say both Putin and Ukrainian President Petro Poroshenko appear to be trying to exploit the intensification of the fighting as a means of influencing the new U.S. administration: Putin could be daring Washington to do something about it; Poroshenko can play up Ukraine’s image as the aggrieved nation.

Small-arms fire and heavier detonations were audible Wednesday throughout the city center. The barrage was indiscriminate; on the outskirts of town, Katya Volkova, 60, was killed by shrapnel from a Grad rocket at 7:30 a.m. as she was out for a walk; her distraught daughter Nadya was kneeling over the body and weeping.

Six Ukrainian soldiers have been killed here since Sunday, and 48 have been wounded, while unconfirmed reports indicated that the separatists suffered heavy losses. The number of civilian casualties is not clear.

The 20,000 people who remain here, out of a prewar population of 35,000, are without heat and water after heavy shelling took out electricity lines and wreaked havoc on the city’s Soviet-era coke plant. It is the largest coke producer in Europe and critical to Ukraine’s steel industry.