Sure! Here's a test case generator for the given problem statement:

```python
import random

def tcgen():
    # Generate random values for H and W within the given constraints
    H = random.randint(1, 500)
    W = random.randint(1, 500)
    
    # Generate the grid with random values for a_ij within the given constraints
    grid = [[random.randint(0, 9) for _ in range(W)] for _ in range(H)]
    
    return H, W, grid

def format_test_case(H, W, grid):
    # Format the test case in the required input format
    test_case = f"{H} {W}\n"
    for row in grid:
        test_case += " ".join(map(str, row)) + "\n"
    return test_case

# Example usage
H, W, grid = tcgen()
test_case = format_test_case(H, W, grid)

print("Generated Test Case:")
print(test_case)
```

### Explanation:

- The `tcgen` function generates the dimensions `H` and `W` for the grid, ensuring they fall within the given constraints (1 ≤ H, W ≤ 500).
- It then generates a grid with random values for `a_{ij}`, where each value falls within the range 0 ≤ `a_{ij}` ≤ 9.
- The `format_test_case` function formats the generated test case into the required input format for the problem.

### Example Output:

Running the script might produce an output similar to:

```
Generated Test Case:
4 4
2 1 9 0
7 4 0 3
6 3 8 7
5 0 1 2
```

The exact output will vary each time the script is executed due to the random nature of the generator.