Continuity of the function $F(y)=\int\limits_0^{+\infty}ye^{-(x-y)^4}dx$

Question

$$F(y)=\int\limits_0^{+\infty}ye^{-(x-y)^4}dx,\text{while } y>0 $$ My attempt: We have $D = [0;+\infty)\times (0;+\infty)$, but let's consider $\overline D = [0;+\infty)\times [\delta;+\infty), \delta > 0$. As $ye^{-(x-y)^4}$ is continuous on $\overline D$, if I prove uniform convergence here $\forall \delta > 0$, I'll get that $F(y)$ is continous on $\overline D$, hence it's continuous on $D$. So my problem is to find majorant 'cause I think it's the problem on Weierstrass test. But I really can't find it. I tried to differentiate but it didn't help much, so I'm completely stuck