We consider a class of stochastic heat equations driven by truncated
α-stable white noises for 1<α<2 with noise coefficients that are
continuous but not necessarily Lipschitz and satisfy globally linear growth
conditions. We prove the existence of weak solution, taking values in two
different spaces, to such an equation using a weak convergence argument on
solutions to the approximating stochastic heat equations. For 1<α<2 the
weak solution is a measure-valued c\`{a}dl\`{a}g process. However, for
1<α<5/3 the weak solution is a c\`{a}dl\`{a}g process taking function
values, and in this case we further show that for 0<p<5/3 the uniform p-th
moment for Lp-norm of the weak solution is finite, and that the weak
solution is uniformly stochastic continuous in Lp sense and satisfies a flow
property